# Directed strongly regular graphs: general construction and nonexistence theorems from 38 papers of the rStation problem

## 1. Problem and contribution

A directed strongly regular graph with parameters $(v, k, t, \lambda, \mu)$ is a loopless directed graph on $v$ vertices whose adjacency matrix $A$ satisfies

$$
AJ = JA = kJ, \qquad A^2 = tI + \lambda A + \mu (J - I - A), \tag{1}
$$

where $J$ is the all-ones matrix, and $0 < t < k$ [1]. Every vertex has $k$ outgoing and $k$ incoming arcs, $t$ of them reciprocal, and the number of paths of length 2 from $x$ to $y \ne x$ is exactly $\lambda$ if $x \to y$ is an arc and exactly $\mu$ if it is not. From now on such a graph is called a DSRG.

The Hobart–Brouwer table [2] lists the feasible sets with $v \le 110$ and marks for each whether a graph has been constructed, nonexistence has been proved or the question is open. A set and its complement are settled together, so from now on we speak of pairs of a set and its complement; a pair is written as the set with $k \le (v - 1)/2$ unless stated otherwise. The problem `strongly_regular_graphs` on rStation [3] asks to settle pairs that are open in the table [2]. Classical constructions, nonexistence conditions and the determinant bounds needed for searches come from the works [4]–[8] (Section 2). On 23 October 2026 the problem had 38 papers [9]–[46]; the survey reads them in the versions given in the references. Each deals with sets that were open in the table when it was published, and the families of the papers also pass through sets constructed earlier. Version 3 of the paper [46] appeared later; its rank-three bordering is treated in Sections 5 and 9.1.

This survey examines these 38 papers. It is meant for whoever takes on the next set. In place of 38 scattered texts it collects the general theorems on which the constructions and obstructions of the papers rest, with proofs, in uniform notation and in a form that applies directly to new sets. The tables of Section 8 set against each claim of a paper its theorem, the paper's evidence and what is checked here. The survey does not judge the state of the table [2] as a whole: it does not count open pairs and does not consider papers outside the problem [3], except those named in Section 2.

**Evidence.** Each claim of a paper about the existence or nonexistence of a graph is described by two attributes. The first is the evidence that the paper itself gives; the classes are listed from strongest to weakest:

- *graph in the text* — the graph is given in the text by an explicit rule or a matrix, and all inputs of the rule are also in the text;
- *proof* — the claim is proved in the text of the paper, but the graph cannot be recovered from the text: the inputs of the rule are external (a net, a Hadamard matrix, a seed from a catalogue or from another paper) or the argument is nonconstructive;
- *graph in the supplement* — the data of the graph or of the seed are in the supplement to the paper;
- *certificate* — the computation comes with a checkable certificate: DRAT, LRAT, pseudo-Boolean (PB), an integer Farkas certificate, a proof in Lean or a recorded list of exclusions that a separate program checks;
- *two programs* — independent programs of the paper gave the same answer, with no certificate;
- *one program* — a computation by one program without a certificate;
- *not described* — the text gives no checkable evidence.

A graph in the text is placed above a proof because it is checked by computation rather than by reading an argument. A family gets its class from the weakest link of its data: graph in the text if the text gives the rule and all inputs and every member can be recovered from the text; proof if the existence of the members is proved but the inputs are external; graph in the supplement if the seed or part of the data is only in the supplement; the class of a program or a certificate if membership rests on a computation. When several papers name a pair, the table row carries the strongest of their evidence. For a computation the class is set by the evidence of the exclusion of cases: in any search a program gives the list of cases and the encoding, and these steps are named in place (Section 6.7).

The second attribute is what is checked here: *proof checked* (step by step or by reduction to a theorem of this survey), *graph built and checked* (the graph is built by the rule of the paper or taken from its text, equalities (1) for it and for its complement are checked by exact integer computation, and the graph itself is in the supplement), *search repeated* (by our own search with integer certificates, Section 6.7), *proved here* (the claim is proved by a theorem of this survey, including in place of the paper's computation) or nothing. This survey did not download the supplements of the papers: claims whose data lie there rest on the paper's evidence. The survey trusts the evidence of the papers that is not repeated here, and its arguments rely on such evidence only where this is stated.

The survey's own computations are finite searches. Each place states what is searched, how many objects there are, what is checked and what the answer is; the programs and data are in the supplement to this article (Section 11).

**Contribution.**

1. The constructions of the 38 papers are reduced to six theorems (Section 4), the obstructions to six (Section 6) and the symmetry exclusions to two (Section 7); all theorems are proved in full. Two lemmas of Section 3, on the quadratic identity and on the sign form, remove the check of row sums from all constructions and define the Hadamard lines: one-parameter series of sets on which the sign constructions work.
2. Generalizations and simplifications of the techniques of the papers.
   - Theorem 3 reduces the sign gluings of the papers [9], [26], [30], [32], [40], [46] to one equation. A block between layers that satisfies the Sylvester equation decomposes uniquely into two channels, and the gluing is a graph exactly when the cross products of the channels vanish (part (a)). In the rank-one case (part (b)) a graph with a signed pair of eigenvectors can be glued with any other graph of the same line that has a signed pair for the same root; the result again has such a pair, and the orders of such graphs on the line form an additive semigroup. All multiple and mixed gluings follow at once. A common sign vector of the summands orthogonal to their pair vectors passes to the gluing, and the gluing can then be switched (Theorem 3(d); a special case is Lemma 4 of the paper [40]).
   - Corollary 2.2 generalizes the lift of Gyürki [47]: the assignment of classes is any balanced one, the seeds may be different graphs with the same parameters, and the partition of the seed passes to the lift.
   - Lemma 3 records, for DSRGs and any eigenvalue, the known criterion for an equitable partition into two classes: the indicator of a set lies in the eigenspace exactly when the number of neighbours in it is constant. It yields the bounds on the size of the set on which the partitions of all gluings rest.
   - Theorem 7 describes the sign-square lines. Bordering of a seed of the same line with $S^2 = q^2 I$ gives on them orders at least $2q^2 + 4$ for side blocks of any rank: the trace count bounds both the seed and the module from below, as in the paper [46] for $q = 5$, and the order of the graph is congruent to 2 modulo 4. With constant sums of the seed, the divisibility of the module by 4 also holds for any rank: it follows from a congruence of orders modulo 4 and not from the sign vectors of rank-one bordering. Corollary 7.1 extends the bound to every Hadamard line: the order is at least $(\beta + 1) + 4\lceil(\beta + 1)/4\rceil$; on the lines $\mathcal H(4, -2, 3)$ and $\mathcal H(1, -5, 9)$ this is 56, and bordering of a seed of the same line is excluded for five of their points that the papers under review do not settle.
   - Theorem 8 on the denominator is stated in integers and in one line closes two rays of multiple sets that the papers had closed by search.
   - Theorem 7(5) describes all sets with $k - t = 1$ and $\mu - \lambda = 1$ through divisors of the number $m^2 + 1$: the complement swaps a divisor and its complementary divisor, and the Hadamard line contains exactly the sets with divisors $((q \mp 1)^2 + 4)/4$. For $v \le 110$ only $(18, 4, 3, 0, 1)$ and its complement remain off the lines.
   - Lemmas 6–8 shorten the search over hyperplanes for $t = \mu$; Lemma 8 complements the closed set of rows of Lemma 3.3 of the paper [13] with an equality that directly defines the search. With them our own search with integer Farkas certificates repeats the nonexistence of 18 pairs of ranks 6 and 7 (Section 6.7).
3. Ten graphs of nine sets are built here and checked by exact computation (Section 4.8): eight by the explicit rules of Theorems 2 and 3 from the papers' seeds, and two by a search for the pairs of Theorem 1(c). Eight of the nine sets are those whose construction the papers claimed, and for these the built graph checks the claim; the two graphs of the ninth, $(144, 52, 26, 20, 18)$, test Corollary 2.2. The graph $(108, 49, 29, 20, 24)$ is an example of the gluing of Theorem 3(b) of two graphs of different orders.
4. The tables of Section 8 list the sets that the papers name individually, the families of the papers, nonexistence results and exclusions of classes of realizations; families include constructions, rays of nonexistence and exclusions for whole families of sets; each row gives the papers, the theorem, the paper's evidence and the check here.
5. For the sets that none of the papers under review settles, Section 9 names the theorem that applies to them and the numbers that bound the search.

## 2. Related work

The definition and the basic necessary conditions are due to Duval [1]. Klin, Munemasa, Muzychuk and Zieschang [4] constructed series of graphs from coherent algebras and proved the nonexistence of the set $(14, 5, 4, 1, 2)$; they also observed that the reciprocal and the one-way parts of the adjacency matrix do not commute. Jørgensen treated the sets with $\mu = \lambda$ [5] and obtained general nonexistence conditions, including the absolute bound [6]. Hobart and Williford [7] derived conditions for graphs with an eigenvalue of small multiplicity. Gyürki [47] introduced the π-join: copies of a graph with an equitable partition of the vertices are joined so that each copy receives arcs into the classes of the partitions of the other copies; the constructions of the paper [23] and Corollary 2.2 rest on it. The table [2] collects these conditions and the known constructions and gives Gyürki's list of all vertex-transitive DSRGs on up to 39 vertices, which we use in Section 7. The search bounds of Section 6 need the largest absolute values of determinants of zero-one matrices of small orders [8].

The problem [3] has been the subject of 38 papers [9]–[46]. They were posted from five accounts, and the author of one paper has not been established; many texts carry no author line, and in the list of references the account is given according to citations in other papers on the problem. Table 1 gives, for each paper, the number of pairs that it names individually, by a set or its complement, by three kinds of claims — construction, nonexistence, exclusion of a class of realizations — and the number of families; several papers may make claims about the same pair. The columns of kinds count pairs, not claims: a paper that excludes three classes for one pair gives one pair. A family is a row with a formula in Tables 4–6: a family of constructions, a ray of nonexistence or an exclusion for a whole family of sets.

**Table 1. Papers on the problem and the number of pairs they make claims about.**

| Paper | Author | Named constructions | Nonexistence claims | Class exclusions | Families |
|---|---|---|---|---|---|
| [9] | mira_loom | 15 | 0 | 0 | 3 |
| [10] | teichmuller_lift | 3 | 0 | 0 | 1 |
| [11] | teichmuller_lift | 8 | 0 | 0 | 2 |
| [12] | mira_loom | 39 | 0 | 0 | 15 |
| [13] | kirkman_loom | 11 | 21 | 0 | 13 |
| [14] | teichmuller_lift | 5 | 0 | 0 | 0 |
| [15] | mira_loom | 0 | 7 | 0 | 6 |
| [16] | kirkman_loom | 32 | 0 | 0 | 9 |
| [17] | mira_loom | 46 | 0 | 0 | 10 |
| [18] | teichmuller_lift | 6 | 0 | 0 | 1 |
| [19] | mira_loom | 1 | 2 | 0 | 2 |
| [20] | mira_loom | 1 | 0 | 0 | 1 |
| [21] | mira_loom | 0 | 3 | 0 | 1 |
| [22] | kirkman_loom | 1 | 0 | 6 | 0 |
| [23] | author not stated | 5 | 0 | 0 | 1 |
| [24] | mira_loom | 3 | 0 | 0 | 5 |
| [25] | orbitwright | 11 | 0 | 9 | 0 |
| [26] | mira_loom | 3 | 0 | 0 | 3 |
| [27] | kirkman_loom | 5 | 0 | 14 | 0 |
| [28] | mira_loom | 0 | 0 | 1 | 0 |
| [29] | kirkman_loom | 3 | 0 | 26 | 1 |
| [30] | mira_loom | 11 | 0 | 0 | 2 |
| [31] | teichmuller_lift | 0 | 0 | 9 | 0 |
| [32] | teichmuller_lift | 1 | 0 | 0 | 1 |
| [33] | orbitwright | 0 | 19 | 0 | 8 |
| [34] | mira_loom | 2 | 0 | 0 | 1 |
| [35] | teichmuller_lift | 2 | 0 | 0 | 0 |
| [36] | mira_loom | 4 | 0 | 0 | 1 |
| [37] | mira_loom | 0 | 0 | 1 | 0 |
| [38] | mira_loom | 3 | 0 | 0 | 1 |
| [39] | teichmuller_lift | 0 | 1 | 0 | 0 |
| [40] | mira_loom | 4 | 0 | 0 | 2 |
| [41] | meleta | 0 | 2 | 0 | 0 |
| [42] | mira_loom | 1 | 0 | 0 | 1 |
| [43] | mira_loom | 0 | 1 | 0 | 1 |
| [44] | mira_loom | 13 | 0 | 0 | 6 |
| [45] | teichmuller_lift | 1 | 0 | 0 | 0 |
| [46] | mira_loom | 24 | 0 | 0 | 4 |

By subject the papers fall into four groups.

- **Constructions from an algebraic form.** The reverse product $A = XY$ of matrices of small rank [16], [17], [20], [24], [44], [10], [11]; gluing of copies of a graph along vertex partitions [12], [13], [20], [23], [42], [44]; sign gluings on the Hadamard line [9], [26], [30], [32], [40], [46]; Kronecker layers [9], [16], [19], [25], [36], [40]; group rings and cosets [13], [14], [17], [18], [22], [25], [27], [29], [35], [38], [42], [45]; switching by a sign vector [12], [30], [34], [40].
- **Nonexistence when $t = \mu$.** The reverse factorization of the graph through its column space and cone certificates close whole rays of multiple sets [13], [15], [19], [21], [33]; the paper [33] formalizes some of these certificates in Lean 4.
- **Nonexistence of small sets.** The nonexistence of $(30, 12, 11, 4, 5)$ is proved independently by computations with certificates in [39] and [41]; the paper [45] checks for this pair only the cycle type $[3^{10}]$ and refers to the known nonexistence. The nonexistence of $(30, 11, 9, 2, 5)$ is proved in the paper [43]; it also shows that on the line $(6d, 2d + 1, 2d - 1, 2, d)$ graphs exist only for $d = 2, 3, 4, 6$. The paper [37] studies sign partitions of the graph $(36, 13, 11, 2, 6)$ as input for a construction of order 72.
- **Symmetry exclusions.** The papers [22], [25], [27], [28], [29], [31], [37] exclude classes of realizations of sets: two-orbit (bi-Cayley) and semiregular actions of abelian groups [22], [27], [29], vertex-transitive and vertex-primitive realizations [25], [29], automorphisms of prime order [28], [31], graphs whose one-way part is a Cayley graph [28], and sign eigenvectors of a given form [37]. Such an exclusion does not settle the pair but narrows the search.

We know of no previous surveys of these papers. We did not search the literature outside the problem [3] separately; we name works known to us where they overlap with the papers under review: the π-join of Gyürki [47], the preprint [48] and the graph $(60, 28, 20, 14, 12)$ of Crnković, Švob and Zubović Žutolija, given in the paper [26] (Section 4.3). The preprint [48] builds graphs of several sets from transitive groups, among them $(72, 25, 15, 6, 10)$, whose construction is also claimed by the paper [12] (Section 4.8).

## 3. Notation and two lemmas

Throughout, $A$ is the adjacency matrix of a DSRG $(v, k, t, \lambda, \mu)$, $\mathbf 1$ is the all-ones vector and $\mathbf 1^\perp$ is its orthogonal complement. The reciprocal part $U = A \circ A^{\mathsf T}$ is a symmetric zero-one matrix with $t$ ones in each row; the one-way part $N = A - U$ has $\kappa = k - t$ ones in each row and each column; $D = A - A^{\mathsf T} = N - N^{\mathsf T}$. Put $c = \mu - \lambda$ and $\sigma = 2\kappa - c$.

From (1) and $AJ = kJ$ it follows that $(A - kI)\bigl(A^2 - (\lambda - \mu)A - (t - \mu)I\bigr) = \mu (A - kI) J = 0$. The roots $r > s$ of the polynomial $x^2 - (\lambda - \mu)x - (t - \mu)$ are the nontrivial eigenvalues of $A$, with multiplicities $f$ and $g$. Then

$$
\lambda = \mu + r + s, \qquad t = \mu - rs, \qquad (k - r)(k - s) = \mu v, \qquad (r - s)^2 = c^2 + 4(t - \mu). \tag{2}
$$

The number $\mu$ is positive: for $\mu = 0$ the equality $k(k + c) = t + (v - 1)\mu$, which follows from $A^2 \mathbf 1 = k^2 \mathbf 1$, would give $t = k(k - \lambda) \ge k$, because $\lambda \le k - 1$. Hence $k \notin \{r, s\}$, and for $r \ne s$ the matrix $A$ is diagonalizable: its minimal polynomial $(x - k)(x - r)(x - s)$ has no multiple roots. For all sets of the table [2] the numbers $r \ne s$ are integers. On $\mathbf 1^\perp$ we have $(A - rI)(A - sI) = 0$.

The complement $\bar A = J - I - A$ is a DSRG with parameters $(v,\ v - k - 1,\ v - 2k + t - 1,\ v - 2k + \mu - 2,\ v - 2k + \lambda)$ [1]; it has the same eigenvectors, with eigenvalues $-1 - s > -1 - r$ of multiplicities $g$ and $f$. A set lies on the *Hadamard line* if $v = 4k - 2\lambda - 2\mu$. On this line one works with the *sign form* $S = J - I - 2A$, with zero on the diagonal and $\pm 1$ off it; put also $H = J - 2A = S + I$. When $t = \mu$ we write $d = \mu - \lambda$; then (1) takes the form $A^2 + dA = \mu J$, the eigenvalues of $A$ are $k$, $0$ and $-d$, the multiplicity of $-d$ is $k/d$, and $\operatorname{rank} A = n = k/d + 1$. We denote by $D(m)$ the largest absolute value of the determinant of a zero-one matrix of order $m$.

**Lemma 1 (regularity from the quadratic identity).** Let $A$ be a zero-one matrix with zero diagonal and $A^2 = (r + s)A - rsI + \mu J$ with $\mu \ne 0$. Then $A\mathbf 1 = A^{\mathsf T}\mathbf 1 = k\mathbf 1$ for some $k$, $A$ is a DSRG with parameters $(v, k, \mu - rs, \mu + r + s, \mu)$ provided $0 < \mu - rs < k$, and $(k - r)(k - s) = \mu v$.

*Proof.* $J = (A^2 - (r + s)A + rsI)/\mu$ is a polynomial in $A$, so $AJ = JA$. In $AJ$ all columns equal the vector of row sums of $A$, and in $JA$ all rows equal the vector of column sums; the equality gives one number $k$ for all the sums. The diagonal of $A^2$ is the number of reciprocal neighbours, $\mu - rs$; off the diagonal we get $\lambda = \mu + r + s$ when $A_{xy} = 1$ and $\mu$ when $A_{xy} = 0$. Finally, $A^2\mathbf 1 = k^2\mathbf 1 = \bigl((r + s)k - rs + \mu v\bigr)\mathbf 1$. ∎

Lemma 1 relieves all constructions of checking the row and column sums: the quadratic identity suffices.

**Lemma 2 (sign form).** For a DSRG and $S = J - I - 2A$,

$$
S^2 = (v - 4k + 2\lambda + 2\mu)J + \alpha S + \beta I, \qquad \alpha = -2(r + s + 1), \qquad \beta = -(2r + 1)(2s + 1). \tag{3}
$$

Hence $S^2 = \alpha S + \beta I$ exactly on the Hadamard line. Conversely, let $S$ of order $N$ be a sign matrix with zero diagonal, $S^2 = \alpha S + \beta I$ and $S\mathbf 1 = S^{\mathsf T}\mathbf 1 = c_0\mathbf 1$. Then $A = (J - I - S)/2$ satisfies $A^2 = \mu J - \tfrac{2 + \alpha}{2}A + \tfrac{\beta - 1 - \alpha}{4}I$ with $k = (N - 1 - c_0)/2$ and $\mu = (N - 2c_0 + \alpha)/4$; if $0 < t < k$, this is a DSRG on the Hadamard line. The number $c_0$ is a root of $x^2 - \alpha x - \beta$, that is, $c_0 \in \{-(2r + 1), -(2s + 1)\}$.

*Proof.* $(J - I)^2 = (v - 2)J + I$, $(J - I)(-2A) + (-2A)(J - I) = -4kJ + 4A$, and $4A^2$ is replaced by (1) in the form $A^2 = (r + s)A - rsI + \mu J$. Substituting $A = (J - I - S)/2$ gives (3); the coefficient of $J$ equals $v - 4k + 4\mu + 2(r + s) = v - 4k + 2\lambda + 2\mu$. The converse is the same expansion for $(J - I - S)^2$ with $JS = SJ = c_0J$; the Hadamard line is checked by substitution. Last: $S^2\mathbf 1 = c_0^2\mathbf 1 = (\alpha c_0 + \beta)\mathbf 1$. ∎

**Hadamard lines.** By Lemma 2 the sign form of a graph on the Hadamard line satisfies $S^2 = \alpha S + \beta I$ and $S\mathbf 1 = c_0\mathbf 1$, where $c_0 = v - 1 - 2k$ is one of the roots $-(2r + 1)$, $-(2s + 1)$. Conversely, the roots $r > s$ and the choice of $c_0$ determine all parameters as functions of the order $N$:

$$
\mathcal H_{r, s, c_0}(N) = \Bigl(N,\ \tfrac{N - 1 - c_0}{2},\ \mu - rs,\ \mu + r + s,\ \mu\Bigr), \qquad \mu = \tfrac{N - 2c_0 + \alpha}{4}.
$$

We call this series of sets the line $\mathcal H(r, s, c_0)$. Integrality of $\mu$ requires $N \equiv 2c_0 - \alpha \equiv 2(r - s) \pmod 4$. For example, the family $\bigl(N, \tfrac{N - 4}{2}, \tfrac{N + 20}{4}, \tfrac{N - 4}{4}, \tfrac{N - 12}{4}\bigr)$ of the paper [30] is the line $\mathcal H(4, -2, 3)$, and the sets $(36, 16, 14, 8, 6)$ and $(40, 18, 15, 9, 7)$ are its points. The complement maps $\mathcal H(r, s, c_0)$ to $\mathcal H(-1 - s, -1 - r, -c_0)$, because the sign form of the complement equals $-S$. The two roots $c_0$ and $\alpha - c_0$ give two lines with the same $r$ and $s$; switching by Theorem 6 along a common sign vector for the eigenvalue $\theta \in \{r, s\}$ of the matrix $A$ (the eigenvalue $-2\theta$ of the matrix $H$) puts the graph on the line with $c_0 = -(2\theta + 1)$, that is, it either leaves the graph on its line or moves it to the other one.

## 4. Constructions

All constructions of the papers reduce to six theorems. The first says that the graph is a product of two rectangular matrices. The second and the third glue the graph from layers: the second along vertex partitions, the third on the Hadamard line along sign eigenvectors. The fourth describes Kronecker layers, the fifth the group form, and the sixth switching. Each theorem is a criterion: it names what has to be checked, and by Lemma 1 nothing else needs to be checked.

### 4.1. The reverse product

**Theorem 1.** (a) Let $X$ be a $v \times m$ matrix, $Y$ an $m \times v$ matrix and

$$
YX = \alpha I_m + ab^{\mathsf T} + K, \qquad Xa = x\mathbf 1, \qquad b^{\mathsf T}Y = y\mathbf 1^{\mathsf T}, \qquad XKY = 0.
$$

Then $P = XY$ satisfies $P^2 = \alpha P + xyJ$. If $A = P - \sigma I$ is a zero-one matrix with zero diagonal and $xy \ne 0$, then $A$ is a DSRG with eigenvalues $r = \alpha - \sigma$, $s = -\sigma$ and $\mu = xy$, provided $0 < t < k$.

(b) Conversely, every DSRG with $r \ne s$ has such a representation with $\sigma = -s$, $m = f + 1$ and $K = 0$: $A - sI = XY$, $YX = (r - s)I + \mu a'b'^{\mathsf T}$, $Xa' = \mathbf 1$, $b'^{\mathsf T}Y = \mathbf 1^{\mathsf T}$. The same holds with $\sigma = -r$ and $m = g + 1$. No $m$ smaller than $1 + \min(f, g)$ occurs.

(c) Let $t = \mu$, $d = \mu - \lambda > 0$ and $n = k/d + 1$. For any $n$ linearly independent columns $C$ of the matrix $A$ ($v \times n$) there exist unique $R$ ($n \times v$) and $\alpha \in \mathbb Q^n$ with

$$
A = CR, \qquad C\alpha = \mathbf 1, \qquad \mathbf 1^{\mathsf T}R = \mathbf 1^{\mathsf T}, \qquad \mathbf 1^{\mathsf T}\alpha = v/k, \qquad RC = \mu\alpha\mathbf 1^{\mathsf T} - dI. \tag{4}
$$

The rows $z_x$ of $C$ are binary vectors of length $n$, the columns $\beta_y$ of $R$ satisfy $z_x^{\mathsf T}\beta_y = A_{xy}$, in particular $z_x^{\mathsf T}\beta_x = 0$, and (4) is written as $\sum_x \beta_x z_x^{\mathsf T} = \mu\alpha\mathbf 1^{\mathsf T} - dI$. Conversely, if binary $z_x$ and rational $\beta_y$ and $\alpha$ satisfy $z_x^{\mathsf T}\beta_y \in \{0, 1\}$, $z_x^{\mathsf T}\beta_x = 0$, $z_x^{\mathsf T}\alpha = 1$, $\mathbf 1^{\mathsf T}\beta_y = 1$ and this identity, then $A = (z_x^{\mathsf T}\beta_y)$ is a DSRG $(v, k, \mu, \mu - d, \mu)$ when $0 < \mu < k$.

*Proof.* (a) $P^2 = X(YX)Y = \alpha XY + (Xa)(b^{\mathsf T}Y) + XKY = \alpha P + xyJ$. Substitute $P = A + \sigma I$: $A^2 = (\alpha - 2\sigma)A + (\alpha\sigma - \sigma^2)I + xyJ$, that is, $r + s = \alpha - 2\sigma$ and $rs = \sigma(\sigma - \alpha)$, whence $\{r, s\} = \{\alpha - \sigma, -\sigma\}$. The rest is Lemma 1.

(b) $A$ is diagonalizable (Section 3), and the eigenvalues of $A - sI$ are $k - s$ once, $r - s$ with multiplicity $f$, and $0$, so $\operatorname{rank}(A - sI) = 1 + f$. Take for $X$ a basis of the column space of $A - sI$; then $A - sI = XY$ with a unique $Y$ of full row rank. On the one hand, $(A - sI)^2 = (A - sI)(A - rI) + (r - s)(A - sI) = \mu J + (r - s)XY$, because $(A - rI)(A - sI) = \mu J$ by (1). On the other, $(A - sI)\mathbf 1 = (k - s)\mathbf 1$ and $\mathbf 1^{\mathsf T}(A - sI) = (k - s)\mathbf 1^{\mathsf T}$, so $\mathbf 1 = Xa'$, $\mathbf 1^{\mathsf T} = b'^{\mathsf T}Y$ and $J = Xa'b'^{\mathsf T}Y$. We get $X\bigl(YX - (r - s)I - \mu a'b'^{\mathsf T}\bigr)Y = 0$; $X$ has a left inverse and $Y$ a right inverse, which gives the equality for $YX$. For $\sigma \notin \{-k, -r, -s\}$ the matrix $A + \sigma I$ is invertible, and for $\sigma = -r$ and $\sigma = -s$ its rank is $1 + g$ and $1 + f$.

(c) The eigenvalues of $A$ are $k$, $0$ and $-d$; the trace $0 = k - dg'$ gives the multiplicity $g' = k/d$ of the eigenvalue $-d$, and $\operatorname{rank} A = n$. The columns of $C$ are a basis of the column space, so $A = CR$ with a unique $R$. The vector $A\mathbf 1 = k\mathbf 1$ lies in the column space, and $C\alpha = \mathbf 1$ for a unique $\alpha$. The column sums of $C$ equal $k$, hence $k\mathbf 1^{\mathsf T}R = \mathbf 1^{\mathsf T}CR = \mathbf 1^{\mathsf T}A = k\mathbf 1^{\mathsf T}$, and $v = \mathbf 1^{\mathsf T}C\alpha = k\mathbf 1^{\mathsf T}\alpha$. Substitute $A = CR$ into $A^2 + dA = \mu J = \mu C\alpha\mathbf 1^{\mathsf T}R$: $C(RC + dI - \mu\alpha\mathbf 1^{\mathsf T})R = 0$, and the full rank of $C$ and $R$ gives (4). Conversely, $A^2 = C(RC)R = \mu(C\alpha)(\mathbf 1^{\mathsf T}R) - dCR = \mu J - dA$, and Lemma 1 with $r = 0$, $s = -d$ gives the set $(v, k, \mu, \mu - d, \mu)$. ∎

Part (c) is part (b) for $t = \mu$: then $r = 0$, $s = -d$, and for $A - rI = A$ we get $m = 1 + g = n$. By the same argument the identity holds for $B = J - A$: $B^2 = uJ + dB$ with $u = v - 2k + \mu - d$, the rank is the same $n$, and the row sums are $v - k$.

**What Theorem 1 covers.** Part (b) makes it a description of all sets: the question is not whether a representation exists but whether one can find $X$ and $Y$ of small rank with a simple structure. When $t = \mu$ this is the factorization $A = CR$ and the graphs from column masks [16], [17], [20], including the whole line $t = \mu = \lambda + 1$ of the paper [17], constructed by an interval projection. The spectral shift and the list of binary vectors of the paper [22] work with a rank-one correction in $YX$. A cancelling correction $K$, $XKY = 0$, gives the reverse product over incidence structures: decompositions of the complete digraph into copies of a small digraph [44] and flags of root systems [24]; the incidence matrix is $X$ itself. Over a group ring, when $X$, $Y$ and $K$ are matrices over $\mathbb Z[G]$, the theorem gives the form $XHY$ of the papers [10], [11]. Part (c) is the basis of all nonexistence proofs for $t = \mu$ in [13], [15], [19], [21], [33] (Section 6).

**Existence of inputs.** Theorem 1 turns the question about a graph into a question about an input — the matrices $X$, $Y$, $K$, a set in a group or a seed — and a family of a paper rests on an argument that the input exists for all parameters of its range. The papers give arguments of three kinds. The first is an explicit input in the text: the masks and the interval projection [16], [17]; then every member is checked by computation. The second is a general argument without an explicit input. In the paper [11] a Cayley graph of the group $\mathbb F_q \rtimes C_m$ of the form $XHY$ is given by a set $Z \subseteq \mathbb F_q$ of $r = (n - 2)/2$ elements with conditions on residue classes of degree $n$; the paper builds $Z$ greedily, and each step is held by the Weil bound for sums of multiplicative characters (Theorem A of the paper). The result: for even $n \ge 4$, a prime power $q \equiv n + 1 \pmod{2n}$, $q \ge 3n + 1$ and $q > n^2 2^{r - 1}(r\sqrt q + r + 1)$ the graph $(mq,\ q + 2m - 3,\ n + 4m - 4,\ n + 2m - 3,\ n + 2)$ with $m = (q - 1)/n$ exists, so for every even $n \ge 4$ there are infinitely many graphs with $\mu = n + 2$. The argument is checked here: the reduction of the conditions on $Z$ to residue classes and the bound for the greedy step are correct, and the form $XHY$ is Theorem 1 over a group ring. The paper [44] relies in the same way on the theorem of Lamken and Wilson on decompositions of the complete digraph for large $a$ (Theorem 7 of the paper), and it does not compute the threshold. The third kind is a finite search: the sets $Z$ of the paper [10] and of Theorem B of the paper [11] are found by a search and rechecked by a separate program. Theorem B closes $n = 4, 6, \ldots, 14$ for all admissible $q$ below the threshold of Theorem A, except $(n, q) = (8, 25)$ and $(14, 43)$: there the exhaustive search found no set, the set $(75, 28, 16, 11, 10)$ is known from other constructions, and the paper does not decide $(129, 46, 22, 17, 16)$. The class of a family in Table 4 follows the kind of argument: an explicit input gives graph in the text, a general argument gives proof, a search gives a program class.

### 4.2. The indicator in an eigenspace and gluing along partitions

**Lemma 3 (indicator).** Let $\theta \in \{r, s\}$, let $F$ be a nonempty proper subset of the vertices, $w = |F|$, and $\mathbf 1_F$ its indicator. The following conditions are equivalent:

1. $\mathbf 1_F - (w/v)\mathbf 1$ is a right eigenvector of $A$ for $\theta$;
2. every vertex $x$ has exactly $\theta[x \in F] + a$ out-neighbours in $F$, for some number $a$.

When they hold, $a = (k - \theta)w/v$ is an integer, $0 \le \theta + a \le \min(k, w - 1)$ and $a \le \min(k, w)$. Therefore $w \ge -\theta v/(k - \theta)$ when $\theta < 0$, and $w \ge (\theta + 1)v/(v - k + \theta)$ when $\theta \ge 0$. The complement $V \setminus F$ satisfies the same conditions with $v - w$ in place of $w$. For left vectors the same holds with $A^{\mathsf T}$ and in-neighbours.

*Proof.* $A\mathbf 1 = k\mathbf 1$, so the equality $A(\mathbf 1_F - (w/v)\mathbf 1) = \theta(\mathbf 1_F - (w/v)\mathbf 1)$ can be rewritten as $A\mathbf 1_F = \theta\mathbf 1_F + \tfrac{(k - \theta)w}{v}\mathbf 1$, and this is condition 2. The coordinate $(A\mathbf 1_F)_x$ is the number of out-neighbours of $x$ in $F$; for a vertex outside $F$ (one exists) it equals $a$, so $a$ is an integer. For a vertex of $F$ it equals $\theta + a$; the vertex itself is not in its own neighbourhood, so $\theta + a \le w - 1$, and it has $k$ neighbours altogether. The inequalities on $w$ come from substituting $a = (k - \theta)w/v$ into $\theta + a \ge 0$ and $\theta + a \le w - 1$. For the complement: $\mathbf 1 - \mathbf 1_F - \tfrac{v - w}{v}\mathbf 1 = -(\mathbf 1_F - \tfrac wv\mathbf 1)$. ∎

**Corollary 3.1 (binary vectors when $t = \mu$).** Let $t = \mu$ and $f \in \{0, 1\}^v$, $f \ne 0$. The vector $f$ lies in the column space of $A$ if and only if $Af = (\mu|f|/k)\mathbf 1 - df$; then $\mu|f|/k$ is an integer and $|f| \ge dk/\mu$. For the row space the same holds with $A^{\mathsf T}$.

*Proof.* If $f = Au$, then $Af = A^2u = \mu(\mathbf 1^{\mathsf T}u)\mathbf 1 - dAu$ and $|f| = \mathbf 1^{\mathsf T}Au = k\mathbf 1^{\mathsf T}u$. Conversely, $f = \bigl((\mu|f|/k)\mathbf 1 - Af\bigr)/d$ and $\mathbf 1 = A\mathbf 1/k$. Integrality follows from the coordinate at $f_x = 0$, and the bound from the nonnegativity of the coordinate at $f_x = 1$. ∎

The column space of $A$ when $t = \mu$ is $\langle\mathbf 1\rangle \oplus E(-d)$, so Corollary 3.1 is Lemma 3 with $\theta = -d$.

**Definition.** A partition $\{C_1, \ldots, C_q\}$ of the vertices of a zero-one matrix $P$ of order $n$ with row and column sums $k$ is called *θ-out* if every class satisfies condition 2 of Lemma 3 with one and the same $\theta$: $P\mathbf 1_{C_a} = \theta\mathbf 1_{C_a} + \alpha\mathbf 1$. A *θ-in* partition is the same for $P^{\mathsf T}$.

Adding up the conditions over the classes gives $k = \theta + q\alpha$; therefore $\alpha = (k - \theta)/q$, and when $\theta \ne k$ all classes have size $n/q$. The trivial partition ($q = 1$) is θ-out for every $\theta$. If a θ-out partition into $q$ classes and a θ′-in partition into $q'$ classes of the same matrix are nontrivial and $\theta \ne \theta'$, they meet uniformly, in $n/(qq')$ vertices: for $u = \mathbf 1_{C_a} - \mathbf 1/q$ and $w = \mathbf 1_{D_b} - \mathbf 1/q'$ we have $w^{\mathsf T}Pu = \theta w^{\mathsf T}u = \theta' w^{\mathsf T}u$, whence $w^{\mathsf T}u = 0$.

**Theorem 2 (gluing layers along partitions).** Let $P_1, \ldots, P_m$ be zero-one matrices with zero diagonal, of orders $n_i$, with row and column sums $k_i$ and $P_i^2 = (r + s)P_i - rsI + \mu_iJ$ for the same $r \ne s$. For each ordered pair $i \ne j$ choose:

- a $\theta_{ij}$-out partition $\mathcal U_{ij}$ of layer $i$ into $q_{ij}$ classes, with class label $a_{ij}(x)$ and $\alpha_{ij} = (k_i - \theta_{ij})/q_{ij}$;
- a $\theta'_{ij}$-in partition $\mathcal V_{ij}$ of layer $j$ into $q'_{ij}$ classes, with label $b_{ij}(y)$, where $\theta'_{ij} = r + s - \theta_{ij}$, and $\beta_{ij} = (k_j - \theta'_{ij})/q'_{ij}$;
- a zero-one matrix $M_{ij}$ of size $q_{ij} \times q'_{ij}$ with row sums $\rho_{ij}(a)$ and column sums $\gamma_{ij}(b)$.

The block $R_{ij}$ from layer $i$ to layer $j$ contains the arc $x \to y$ exactly when $M_{ij}[a_{ij}(x), b_{ij}(y)] = 1$. Suppose the **intersection condition** holds: for every $h$ and $i, j \ne h$ the partitions $\mathcal V_{ih}$ and $\mathcal U_{hj}$ of layer $h$ meet uniformly, in $c_{ihj} = n_h/(q'_{ih}q_{hj})$ vertices. Then the matrix $A$ with diagonal blocks $P_i$ and blocks $R_{ij}$ satisfies $A^2 = (r + s)A - rsI + \mu J$ if and only if, pointwise,

$$
\alpha_{ij}\gamma_{ij}(b_{ij}(y)) + \beta_{ij}\rho_{ij}(a_{ij}(x)) + \sum_{h \ne i, j} c_{ihj}\,\rho_{ih}(a_{ih}(x))\,\gamma_{hj}(b_{hj}(y)) = \mu \tag{5}
$$

for all $i \ne j$, $x$ in layer $i$ and $y$ in layer $j$, and

$$
\mu_i + \sum_{h \ne i} c_{ihi}\,\rho_{ih}(a_{ih}(x))\,\gamma_{hi}(b_{hi}(y)) = \mu \tag{6}
$$

for all $i$ and $x, y$ in layer $i$. The intersection condition holds automatically if $\theta'_{ih} \ne \theta_{hj}$ or one of the two partitions is trivial.

*Proof.* By the definition of the partitions, $(P_iR_{ij})(x, y) = \sum_a |N^+(x) \cap C_a|\,M_{ij}[a, b_{ij}(y)] = \theta_{ij}R_{ij}(x, y) + \alpha_{ij}\gamma_{ij}(b_{ij}(y))$, where $N^+(x)$ are the out-neighbours of $x$ in layer $i$ and $C_a$ are the classes of $\mathcal U_{ij}$; likewise $(R_{ij}P_j)(x, y) = \theta'_{ij}R_{ij}(x, y) + \beta_{ij}\rho_{ij}(a_{ij}(x))$. By the intersection condition, $(R_{ih}R_{hj})(x, y) = \sum_{b, a'} |D_b \cap C_{a'}|\,M_{ih}[a_{ih}(x), b]\,M_{hj}[a', b_{hj}(y)] = c_{ihj}\rho_{ih}(a_{ih}(x))\gamma_{hj}(b_{hj}(y))$, where $D_b$ are the classes of $\mathcal V_{ih}$ and $C_{a'}$ are the classes of $\mathcal U_{hj}$. The block $(i, j)$ of the matrix $A^2 - (r + s)A + rsI$ for $i \ne j$ equals $(\theta_{ij} + \theta'_{ij} - r - s)R_{ij}$, that is, zero, plus the left-hand side of (5); the block $(i, i)$ equals $\mu_iJ + \sum_{h \ne i}R_{ih}R_{hi}$. Comparison with $\mu J$ gives (5) and (6). ∎

**Corollary 2.1 (constant sums).** Suppose all $M_{ij}$ have constant row sums $\ell_{ij}$ and constant column sums, and the intersection condition holds. Put $Q_{ii} = k_i$, $Q_{ij} = \ell_{ij}n_j/q'_{ij}$, the quotient matrix over the layers. Then $A^2 = (r + s)A - rsI + \mu J$ is equivalent to

$$
Q^2 = (r + s)Q - rsI + \mu\,\mathbf 1\mathbf n^{\mathsf T}, \qquad \mathbf n = (n_1, \ldots, n_m). \tag{7}
$$

*Proof.* With constant sums the left-hand sides of (5) and (6) are constant on a block; denote them by $E_{ij}$. The block $(i, j)$ of $A^2$ equals $(r + s)R_{ij} + E_{ij}J$ (for $i = j$ also $-rsI$). The partition into layers is equitable with quotient $Q$: on the vertices of layer $h$ the vector $A\mathbf 1_{(j)}$ equals $Q_{hj}$, where $\mathbf 1_{(j)}$ is the indicator of layer $j$. Therefore on layer $i$ the vector $A^2\mathbf 1_{(j)}$ equals $(Q^2)_{ij} = (r + s)Q_{ij} - rs[i = j] + E_{ij}n_j$, and $E_{ij} = \mu$ for all $i, j$ is equivalent to (7). ∎

For $m$ identical layers $(n, k, t, \lambda, \mu)$ with a common off-diagonal number $\eta$ the quotient matrix equals $(k - \eta)I + \eta J$, and (7) splits into two equalities whose difference gives $(k - \eta - r)(k - \eta - s) = 0$. Hence **in a gluing of identical layers with constant sums, $\eta \in \{k - r, k - s\}$**, the result is a DSRG $(mn,\ k + (m - 1)\eta,\ \ldots)$ with $\mu' = \mu + (m - 1)\eta^2/n$, and the divisibility $n \mid (m - 1)\eta^2$ is needed. When $t = \mu$ this is $\eta \in \{k, k + d\}$, the layerwise lift of the paper [20]. The gluing along opposite equitable partitions of the paper [12] is Corollary 2.1 in general form.

Condition (7) is only necessary for the existence of a graph with the given set: it speaks of the quotient, not of the partitions. Eleven pairs satisfy it on seeds of two layers — $(24, 10, 5, 3, 5)$, $(32, 11, 6, 5, 3)$, $(36, 14, 7, 6, 5)$, $(48, 20, 10, 6, 10)$, $(64, 23, 13, 12, 6)$, $(72, 21, 7, 4, 7)$, $(72, 29, 15, 14, 10)$, $(72, 30, 15, 9, 15)$, $(96, 35, 20, 19, 9)$, $(96, 40, 20, 12, 20)$ and $(108, 44, 23, 22, 15)$ with their complements — yet graphs with these sets do not exist by the evidence of the papers (Table 5): for seven pairs the search is repeated here, three rest on the certificate [21], and one on two programs [15]. Hence the seeds do not have the partitions that the gluing requires. The seeds and quotients for each of the eleven pairs are listed in the supplement.

**Corollary 2.2 (Gyürki's lift in general form).** Let $P_1, \ldots, P_m$ be DSRGs with the same parameters $(n, k, t, \lambda, \mu)$ and roots $r, s$, let $\theta' \in \{r, s\}$, let the number $c = (k - \theta')/\mu$ be an integer at least 2, and let each $P_i$ have a θ′-in partition into $c$ classes $D^{(i)}_1, \ldots, D^{(i)}_c$, that is, every vertex has $\mu$ incoming arcs from each class other than its own. Let $m = cu + 1$, and let the assignment $\tau(h, i) \in \{1, \ldots, c\}$ for $h \ne i$ be balanced: every layer $i$ receives each class from exactly $u$ layers $h$. Then the block matrix with diagonal blocks $P_i$ and blocks $R_{hi} = \mathbf 1\,\mathbf 1_{D^{(i)}_{\tau(h, i)}}^{\mathsf T}$ is a DSRG

$$
(mn,\ k + un,\ t + ub,\ \lambda + ub,\ \mu + ub), \qquad b = n/c,
$$

with the same $r, s$; the second root $\theta'' = r + s - \theta'$ satisfies $k - \theta'' = b$. The classes $E_a = \bigcup_i D^{(i)}_a$ form a θ′-in partition of the lift into the same $c$ classes.

*Proof.* This is Theorem 2 with trivial out-partitions ($q = 1$, $\theta = \theta''$, $\alpha = k - \theta''$) and $M_{hi} = e_{\tau(h, i)}^{\mathsf T}$, so that $\rho = 1$ and $\gamma(a) = [a = \tau(h, i)]$; the layers of Theorem 2 need not coincide. The intersection condition holds because the out-partitions are trivial; $\beta = (k - \theta')/c = \mu$ and $c_{ihj} = b$. Equality (6): $\mu + b\cdot\#\{h \ne i : \tau(h, i) = a\} = \mu + ub$ for $y \in D^{(i)}_a$, which is constant. Equality (5): for $y \in D^{(j)}_a$ the number of $h \notin \{i, j\}$ with $\tau(h, j) = a$ equals $u - [\tau(i, j) = a]$, and (5) reduces to $(k - \theta'' - b)[\tau(i, j) = a] = 0$. This holds: $(k - \theta')(k - \theta'') = \mu n$ by (2) and $k - \theta' = c\mu$ give $k - \theta'' = n/c$. The number of neighbours of a vertex is $k + (m - 1)b = k + un$.

Partition of the lift. Let $y \in D^{(j)}_{a'}$. From layer $j$, $\theta'[a = a'] + \mu$ arcs from $D^{(j)}_a$ enter $y$. A layer $h \ne j$ sends arcs into $y$ from all its vertices if $\tau(h, j) = a'$, and none otherwise; there are $u$ such layers, and each gives $|D^{(h)}_a| = b$ arcs from $E_a$. In total this is $\theta'[a = a'] + \mu + ub$, which is the condition of a θ′-in partition of the lift with $\mu + ub$ in place of $\mu$. ∎

For identical layers and the cyclic assignment $\tau(h, i) = \lceil ((i - h) \bmod m)/u \rceil$ this is Gyürki's π-join [47, Theorem 4]. Its condition on the partition, the column quotient $(\lambda + b - k)I + \mu(J - I)$, is precisely the condition of a θ′-in partition, because $\lambda + b - k = \mu + \theta'$. The papers [13] and [23] use it to build graphs with $c = 2$, and the papers [42] and [44] with $c$ classes and the cyclic assignment on $\mathbb Z_m$.

The corollary adds three things to [47]: the assignment is any balanced one, the layers are any graphs of one set, each with its own partition, and the partition passes to the lift. These freedoms give no new sets: a repeated lift leads to the set of a lift with $u + u' + cuu'$, because $(cu + 1)(cu' + 1) = c(u + u' + cuu') + 1$. They change the graphs themselves and are needed when the graph of the lift must have one more property, for example a signed pair for the gluings of Section 4.3.

### 4.3. Sign gluing on the Hadamard line

Let $\alpha$, $\beta$ be the numbers of Lemma 2, and $\theta$ and $\bar\theta = \alpha - \theta$ the roots of $x^2 - \alpha x - \beta$.

**Theorem 3 (sign gluing).** (a) Let $S_1, \ldots, S_m$ be sign matrices with zero diagonal, of orders $n_i$, $S_i^2 = \alpha S_i + \beta I$ with $\theta \ne \bar\theta$, and for $i \ne j$ let the blocks $S_{ij}$ be $n_i \times n_j$ sign matrices satisfying the Sylvester equation

$$
S_iS_{ij} + S_{ij}S_j = \alpha S_{ij}. \tag{8}
$$

Then each block decomposes uniquely into the sum of two *channels* $S_{ij} = S^+_{ij} + S^-_{ij}$ with

$$
S_iS^+_{ij} = \theta S^+_{ij}, \quad S^+_{ij}S_j = \bar\theta S^+_{ij}, \qquad S_iS^-_{ij} = \bar\theta S^-_{ij}, \quad S^-_{ij}S_j = \theta S^-_{ij},
$$

namely $S^+_{ij} = (S_i - \bar\theta I)S_{ij}/(\theta - \bar\theta)$. The products of like channels $S^+_{ih}S^+_{hj}$ and $S^-_{ih}S^-_{hj}$ vanish, and the matrix $S$ with blocks $S_i$ and $S_{ij}$ satisfies $S^2 = \alpha S + \beta I$ if and only if

$$
\sum_{h \notin \{i, j\}} S^+_{ih}S^-_{hj} = 0 \quad \text{and} \quad \sum_{h \notin \{i, j\}} S^-_{ih}S^+_{hj} = 0 \quad \text{for all } i, j, \text{ including } i = j. \tag{9}
$$

For $m = 2$ equation (8) is not a hypothesis but a consequence of $S^2 = \alpha S + \beta I$. If moreover $S\mathbf 1 = S^{\mathsf T}\mathbf 1 = c_0\mathbf 1$, then $(J - I - S)/2$ is a DSRG on the Hadamard line when $0 < t < k$ (Lemma 2). A *one-channel* block is a block with $S_iS_{ij} = \theta_{ij}S_{ij}$ and $S_{ij}S_j = \bar\theta_{ij}S_{ij}$, $\theta_{ij} \in \{\theta, \bar\theta\}$; if all blocks are of this kind, condition (9) says that $\sum_h S_{ih}S_{hj} = 0$ over those $h$ with $\theta_{hj} \ne \theta_{ih}$. For channels of rank one, $S^+_{ih} = c_ia_h^{\mathsf T}$ and $S^-_{hj} = d_hb_j^{\mathsf T}$ with vectors common to all partners of a layer, the cross product equals $(a_h^{\mathsf T}d_h)\,c_ib_j^{\mathsf T}$: condition (9) holds if in each layer $a_h \perp d_h$ and $b_h \perp c_h$, that is, the left and right vectors of one value are orthogonal.

(b) *Rank-one gluing.* Let $A_1, \ldots, A_m$ be graphs of one line $\mathcal H(r, s, c_0)$ of orders $n_i$ with $0 < t_i \le k_i$ (undirected graphs with $t_i = k_i$ are allowed for $\mu_i > 0$), let $\theta \in \{r, s\}$ and let $\theta'$ be the other root. Suppose each $A_i$ has a *signed pair for $\theta$*: a right eigenvector $x_i \in \{\pm 1\}^{n_i}$ for $\theta$ and a left eigenvector $y_i \in \{\pm 1\}^{n_i}$ for $\theta'$. Then for $m \ge 2$ the matrix $A$ with diagonal blocks $A_i$ and blocks $R_{ij} = \tfrac12(J + x_iy_j^{\mathsf T})$ for $i \ne j$ is a DSRG $\mathcal H_{r, s, c_0}(n_1 + \cdots + n_m)$, and the vectors $(x_1, \ldots, x_m)$ and $(y_1, \ldots, y_m)$ are its signed pair for $\theta$. Only graphs of order divisible by 4 can have a signed pair, and therefore only on lines with even $r - s$.

(c) *Semigroup of orders.* The orders of graphs of the line $\mathcal H(r, s, c_0)$ with a signed pair for $\theta$ are closed under addition. In particular, one such graph of order $n$ gives graphs of all orders $mn$.

(d) *Common vector of the gluing.* Suppose that in part (b) each $A_i$ has a sign vector $z_i$ that is a right and a left eigenvector for $\theta$. The vector $(z_1, \ldots, z_m)$ is a common right and left eigenvector of the gluing for $\theta$ if and only if $z_i^{\mathsf T}x_i = 0$ for all $i$; then the gluing can be switched by Theorem 6. For common vectors $z_i$ of the root $\theta'$ the condition is the same with $y_i$ in place of $x_i$.

*Proof.* (a) From $S_i^2 = \alpha S_i + \beta I$ it follows that $(S_i - \theta I)(S_i - \bar\theta I) = 0$, and for $\theta \ne \bar\theta$ the matrix $E_i = (S_i - \bar\theta I)/(\theta - \bar\theta)$ is a projector with $S_iE_i = \theta E_i$ and $S_i(I - E_i) = \bar\theta(I - E_i)$. By (8), $S_iS_{ij} = S_{ij}(\alpha I - S_j)$, so $E_iS_{ij} = S_{ij}(\theta I - S_j)/(\theta - \bar\theta) = S_{ij}(I - E_j)$. Hence the channels $S^+_{ij} = E_iS_{ij} = S_{ij}(I - E_j)$ and $S^-_{ij} = (I - E_i)S_{ij} = S_{ij}E_j$ satisfy the four equalities. The decomposition is unique: any channel $S^+$ with these equalities lies in the image of $E_i$, and $S^-$ lies in the image of $I - E_i$. Conversely, the sum of such channels satisfies (8), because $\theta + \bar\theta = \alpha$. The block $(i, j)$ of the square for $i \ne j$ equals $S_iS_{ij} + S_{ij}S_j + \sum_{h \notin \{i, j\}}S_{ih}S_{hj} = \alpha S_{ij} + \sum$, and the block $(i, i)$ equals $\alpha S_i + \beta I + \sum_{h \ne i}S_{ih}S_{hi}$; therefore $S^2 = \alpha S + \beta I$ is equivalent to $\sum_{h \notin \{i, j\}}S_{ih}S_{hj} = 0$ for all $i, j$. Like channels annihilate each other: $S^+_{ih}S^+_{hj} = S_{ih}(I - E_h)E_hS_{hj} = 0$ and $S^-_{ih}S^-_{hj} = S_{ih}E_h(I - E_h)S_{hj} = 0$. Therefore $S_{ih}S_{hj} = S^+_{ih}S^-_{hj} + S^-_{ih}S^+_{hj}$, where the first term equals $E_i(S_{ih}S_{hj})E_j$ and the second equals $(I - E_i)(S_{ih}S_{hj})(I - E_j)$. A sum $P + Q$ with $P = E_iPE_j$ and $Q = (I - E_i)Q(I - E_j)$ vanishes exactly when $P = Q = 0$: multiplication by $E_i$ on the left and by $E_j$ on the right leaves $P$ of it. This gives (9). For $m = 2$ the sum over $h \notin \{1, 2\}$ is empty, and the block $(1, 2)$ of the square equals $S_1S_{12} + S_{12}S_2$; the equality $S^2 = \alpha S + \beta I$ gives (8).

(b) The values $\theta$, $\theta'$ differ from $k_i$ (Section 3), so $\mathbf 1^{\mathsf T}x_i = \mathbf 1^{\mathsf T}y_i = 0$, and $y_i^{\mathsf T}A_ix_i$ equals both $\theta y_i^{\mathsf T}x_i$ and $\theta' y_i^{\mathsf T}x_i$, whence $y_i^{\mathsf T}x_i = 0$. Put $S_i = J - I - 2A_i$ and $S_{ij} = J - 2R_{ij} = -x_iy_j^{\mathsf T}$. Then $S_ix_i = -(2\theta + 1)x_i$ and $y_j^{\mathsf T}S_j = -(2\theta' + 1)y_j^{\mathsf T}$, so $S_iS_{ij} = -(2\theta + 1)S_{ij}$ and $S_{ij}S_j = -(2\theta' + 1)S_{ij}$, and the sum of these numbers is $-2(r + s) - 2 = \alpha$: the blocks satisfy (8), and each of them has one channel with the same pair of roots. Condition (9) holds: $S_{ih}S_{hj} = x_i(y_h^{\mathsf T}x_h)y_j^{\mathsf T} = 0$. The blocks $S_{ij}$ have zero row and column sums, so $S\mathbf 1 = S^{\mathsf T}\mathbf 1 = c_0\mathbf 1$ with the $c_0$ common to the line. By part (a) and Lemma 2, $A = (J - I - S)/2$ is a graph on the line $\mathcal H(r, s, c_0)$ of order $N = \sum n_i$. On the line, $t = \mu - rs$ and $k - t = (N - 2 - \alpha)/4 + rs$ grow with $N$, so $0 < t < k$ for $m \ge 2$. The signed pair: block $i$ of the vector $A(x_1, \ldots, x_m)$ equals $A_ix_i + \tfrac12\sum_{j \ne i}(J x_j + x_i\,y_j^{\mathsf T}x_j) = \theta x_i$, and likewise for $y$. Finally, $\mathbf 1$, $x_i$ and $y_i$ are pairwise orthogonal sign vectors of length $n_i$. In the coordinates where $x_i$ and $y_i$ take the signs $(\varepsilon, \varepsilon')$, the four patterns occur $a_{++}, a_{+-}, a_{-+}, a_{--}$ times; the three orthogonality conditions and the sum $n_i$ give $a_{\varepsilon\varepsilon'} = n_i/4$. On the line $N \equiv 2(r - s) \pmod 4$ (Section 3), so $r - s$ is even.

(c) This is part (b) with $m = 2$, and with $m$ copies of one graph.

(d) Let $z = (z_1, \ldots, z_m)$. Block $i$ of the vector $Az$ equals $\theta z_i + \tfrac12\sum_{j \ne i}(Jz_j + x_i\,y_j^{\mathsf T}z_j) = \theta z_i$: $\mathbf 1^{\mathsf T}z_j = 0$, and $y_j^{\mathsf T}z_j = 0$ because $y_j$ is a left vector for $\theta'$ and $z_j$ a right one for $\theta \ne \theta'$. Block $j$ of the vector $z^{\mathsf T}A$ equals $\theta z_j^{\mathsf T} + \tfrac12\bigl(\sum_{i \ne j}z_i^{\mathsf T}x_i\bigr)y_j^{\mathsf T}$, and it equals $\theta z_j^{\mathsf T}$ for all $j$ exactly when all the sums $\sum_{i \ne j}z_i^{\mathsf T}x_i$ vanish, that is, when all $z_i^{\mathsf T}x_i = 0$ (for $m \ge 2$). A common vector of $A$ for $\theta$ is orthogonal to $\mathbf 1$ and is therefore a common vector of $H = J - 2A$ for $-2\theta$, as Theorem 6 requires. For the root $\theta'$ the roles of $x_i$ and $y_i$ are exchanged. ∎

Part (d) selects the seeds whose gluing can be switched. The graph $(36, 16, 14, 8, 6)$ has eight right sign vectors for 4, all of them common, but their pairwise inner products equal $\pm 12$ and $\pm 36$; none is orthogonal to the pair vector $x$, and the gluings of its copies have no common vector of the form $(z_1, \ldots, z_m)$ for 4. For the root $-2$ the condition of part (d) is orthogonality to the vector $y$, but switching by a common vector for $-2$ gives $c_0 = 3$ and leaves the graph on the line $\mathcal H(4, -2, 3)$.

Part (b) also gives a filter. If the set $\mathcal H_{r, s, c_0}(n_1 + n_2)$ does not exist, then no pair of graphs of the sets $\mathcal H_{r, s, c_0}(n_1)$ and $\mathcal H_{r, s, c_0}(n_2)$ has signed pairs for the same $\theta$. The converse is false: the absence of a signed pair in the known graphs of a set says nothing about the existence of the sum.

**Lemma 4 (compensation).** If $S_i^2 = \alpha S_i + \beta I$, $S_ix = \theta x$ and $S_i\mathbf 1 = c_0\mathbf 1 + \tau x$ with $\tau \ne 0$ and $x$ not proportional to $\mathbf 1$, then $c_0 = \bar\theta$.

*Proof.* $S_i^2\mathbf 1 = c_0^2\mathbf 1 + \tau(c_0 + \theta)x$, while $\alpha S_i\mathbf 1 + \beta\mathbf 1 = (\alpha c_0 + \beta)\mathbf 1 + \alpha\tau x$; comparing the coefficients of $x$ gives $c_0 + \theta = \alpha$. ∎

Lemma 4 describes blocks with nonconstant row sums that are levelled out by the contribution of the neighbouring blocks: the "base" sum of such a block must equal the paired root. The bordering of sign squares in the paper [46] rests on this: a seed with $S^2 = q^2I$ is bordered by a module $T$ with $T^2 = q^2I$ and rank-one channels.

**What Theorem 3 covers.** The sign-channel composition of the paper [32] is part (a) with one-channel blocks of rank one $c_ia_j^{\mathsf T}$ or $d_ib_j^{\mathsf T}$ for any number of layers, where the channel is chosen on each pair of layers: its conditions $b_h^{\mathsf T}c_h = a_h^{\mathsf T}d_h = 0$ are condition (9), and its lemma on the orders of seeds being multiples of 4 is the case $m = 2$ of the congruence from the paragraph "Restrictions". The sign gluing of Lemma 4 of the paper [40] is part (b) for any number of layers: $a$ copies of a seed of order 36 and $b$ copies of a seed of order 48 are glued by the blocks $d_ib_j^{\mathsf T}$ of a common signed pair, and the switching by the glued common vector in the same paper is a special case of part (d). The paper [37] calls the two-copy gluing a special case of the composition of sign channels of the papers [32] and [26] and gives no gluing theorem of its own. The sign gluing of the paper [26] is part (a) with one-channel blocks of rank one: to a seed of order 28, a switching of the graph $T(8)$, $m$ copies of the graph $(60, 28, 20, 14, 12)$ are attached by the blocks $ub^{\mathsf T}$, $av^{\mathsf T}$ and $ac^{\mathsf T}$, and condition (9) is held by the equalities $v^{\mathsf T}u = b^{\mathsf T}a = c^{\mathsf T}a = 0$. The pairs of vectors differ between different pairs of layers, so this is not part (b). The channel gluing of the paper [30] is part (a) with blocks of two channels of rank one: $S_{ij} = c_ia_j^{\mathsf T} + d_ib_j^{\mathsf T}$, where $c_i$ and $d_i$ are right eigenvectors of $S_i$ for its two values with disjoint supports and entries in $\{0, \pm 1\}$, $a_j$ and $b_j$ are left sign eigenvectors, and the vectors of a layer are common to all its partners. The paper's conditions $a_i^{\mathsf T}d_i = b_i^{\mathsf T}c_i = 0$ are condition (9) in the form $a_h \perp d_h$, $b_h \perp c_h$. Ternary vectors are needed because the seed of order 40 has no right sign eigenvector for 4. In this way the paper glues the line $\mathcal H(4, -2, 3)$ from seeds of orders 36, 40, 48, 60 and 64, and it assembles the seed of order 64 itself from graphs of orders 28 and 36 by blocks of ranks 2 and 5. Part (b) is the case of this gluing in which all layers have signed pairs for one root and the second channel is empty. The bordering [46] is part (a) with Lemma 4. The sign extension of the paper [9] is part (a) with $m = 2$ in the form $H = S + I$: the layers $H$ and $D$ are glued by the one-channel blocks $hb^{\mathsf T}$ and $az^{\mathsf T}$, where $h$ is a column of $H$; for $t = \mu$ on the Hadamard line $\beta = \alpha + 1$, so $H^2 = 2dH$, and every column of $H$ is a right eigenvector for $2d$.

**Restrictions.** Gluing $m$ seeds of one line by blocks of any rank requires $(m - 1)(r - s)$ to be even: the orders of the seeds and of the result are congruent to $2(r - s)$ modulo 4. Theorem 3(a) restricts one-channel blocks by the sign vectors of the seeds with constant row and column sums $c_0$, for any rank of the block. Read by columns and rows, the equalities of a one-channel block say: every column of a block $S_{ij}$ is a right sign eigenvector of $S_i$ for $\theta_{ij}$, and every row is a left sign eigenvector of $S_j$ for $\bar\theta_{ij}$. If the value $\theta_{ij}$ differs from $c_0$, such a column is orthogonal to $\mathbf 1$, because $\mathbf 1^{\mathsf T}S_i = c_0\mathbf 1^{\mathsf T}$, and is therefore a sign eigenvector of $A_i$ for the root $-(\theta_{ij} + 1)/2$; the same holds for rows, because $S_j\mathbf 1 = c_0\mathbf 1$. For the value $c_0$ the vector $\mathbf 1$ itself qualifies. Hence the sign vectors of the seeds on the right and on the left for the root with $-(2\theta + 1) \ne c_0$ decide, before any search, which directions of one-channel blocks are possible (Section 9.6 gives an example). For a block of two channels the columns are sums of eigenvectors for two values, and this filter does not apply to it: this is how the seed of order 40 without a right sign vector for 4 enters the gluings of the paper [30].

**Example: the line $\mathcal H(4, -2, 3)$.** The graph $(36, 16, 14, 8, 6)$ of the papers [12], [13], [25] lies on this line: $r = 4$, $s = -2$, $\alpha = -6$, $c_0 = 3$. It has a signed pair for $\theta = 4$, a right sign vector for 4 and a left one for $-2$, both in the supplement. By Theorem 3(c), $m$ copies of it give $\mathcal H_{4, -2, 3}(36m) = (36m, 18m - 2, 9m + 5, 9m - 1, 9m - 3)$ for all $m$; for $m = 2, 3$ these are $(72, 34, 23, 17, 15)$ and $(108, 52, 32, 26, 24)$ (Section 4.8).

**Example: summands of different orders.** The line $\mathcal H(1, -5, 9)$ contains the graph $(48, 19, 14, 5, 9)$ of the paper [25] and the graph $(60, 25, 17, 8, 12)$, the complement of the switching of Section 4.6 of the graph $(60, 28, 20, 14, 12)$ of Crnković, Švob and Zubović Žutolija, given in the paper [26]. Both have a signed pair for $\theta = 1$, and Theorem 3(b) glues them into $\mathcal H_{1, -5, 9}(108) = (108, 49, 29, 20, 24)$ (Section 4.8). The order of neither seed divides 108, so copies of a single seed do not suffice: here the semigroup works precisely by adding different orders.

### 4.4. Kronecker layers

**Theorem 4.** Let $X$, $Y$ be zero-one matrices of order $n$, $\operatorname{diag} X = 0$, $m \ge 2$, $A = I_m \otimes X + (J_m - I_m) \otimes Y$, $N' = X - Y$, $P = X + (m - 1)Y$. Then $A^2 = (r + s)A - rsI + \mu J$ is equivalent to

$$
(N' - rI)(N' - sI) = 0 \quad\text{and}\quad (P - rI)(P - sI) = m\mu J.
$$

*Proof.* $A = I \otimes N' + J \otimes Y$ and $J_m^2 = mJ_m$, so $A^2 - (r + s)A + rsI = I \otimes Z + J \otimes W$ with $Z = (N' - r)(N' - s)$ and $W = N'Y + YN' + mY^2 - (r + s)Y$. Equality with $\mu J_m \otimes J_n$ for $m \ge 2$ means $Z = 0$ and $W = \mu J$, and $(P - r)(P - s)$ with $P = N' + mY$ expands into $Z + mW$. ∎

Commutativity of $X$ and $Y$ is not needed. The blow-up $A \otimes J_c$ for $t = \mu$ of the paper [16] in the notation of Theorem 4 is the case $X = Y = A$: $N' = 0$, $P = cA$, roots $0$ and $-cd$, result $c\cdot(v, k, \mu, \lambda, \mu)$; hence, when $t = \mu$, a constructed set gives all its multiples. The layer criterion of the paper [36] is Theorem 4 itself; the twin blow-up [25] and the paired sectors [40] are special cases of it; the multiples of the papers [9] and [19] are obtained in the same way.

### 4.5. The group form

**Theorem 5.** Let a group $G$ act semiregularly on the vertices with $m$ orbits, so that $A$ is a matrix $F \in M_m(\mathbb Z[G])$ with zero-one coefficients, and let there be no loops: in the diagonal entries of $F$ the coefficient of the identity of the group is zero. Then $A$ is a DSRG with roots $r, s$ and the given $\mu$ if and only if (a) the quotient $\varepsilon(F)$ under the trivial character satisfies $\varepsilon(F)^2 = (r + s)\varepsilon(F) - rsI + \mu|G|J_m$ and (b) $(\rho(F) - rI)(\rho(F) - sI) = 0$ for every nontrivial irreducible representation $\rho$ of the group $G$. As in Lemma 1, one also needs $0 < \mu - rs < k$.

*Proof.* The DSRG identity in $M_m(\mathbb Z[G])$ is $F^2 = (r + s)F - rsI + \mu\Omega$, where all entries of $\Omega$ equal $\sum_{g \in G}g$. The regular representation is faithful and decomposes into the sum of all irreducible ones, so the identity is equivalent to its images under all $\rho$. For $\rho = \varepsilon$ the image of $\Omega$ is $|G|J_m$, and for nontrivial $\rho$ the image of $\sum_g g$ is zero by Schur orthogonality. ∎

The version for cosets of a subgroup is the same with the irreducible constituents of the permutation representation. Theorem 5 covers the constructions on groups in the papers [13], [14], [17], [18], [22], [25], [27], [29], [35], [38], [42], [45]: Cayley and bi-Cayley graphs, two-layer and three-orbit forms, the affine construction on cosets of $\mathbb F_q \rtimes H$ and the character sieves; each paper uses part of these techniques. This includes the graph $(54, 23, 22, 9, 10)$ of the paper [45], the three-orbit form of the paper [35] and three pairs of order 100 of the paper [29].

### 4.6. Switching

**Theorem 6.** Let $A$ be a DSRG on the Hadamard line, $H = J - 2A$, and let the sign vector $z$ be a common right and left eigenvector of $H$ for $\theta$. Then $H' = D_zHD_z$, where $D_z = \operatorname{diag} z$, defines a DSRG $A' = (J - H')/2$ with the same $r, s$, degree $k' = (v - \theta)/2$ and $\mu' = \bigl(v - 2\theta - 2(\lambda - \mu)\bigr)/4$, provided $0 < t' < k'$.

*Proof.* $H = S + I$, and by Lemma 2 $H^2 = (\alpha + 2)H + (\beta - \alpha - 1)I$. Conjugation by $D_z$ preserves this identity and the unit diagonal, while $H'\mathbf 1 = D_zHz = \theta\mathbf 1$ and $\mathbf 1^{\mathsf T}H' = \theta\mathbf 1^{\mathsf T}$. The converse part of Lemma 2 with $S' = H' - I$ and $c_0 = \theta - 1$ gives a DSRG with $k' = (v - \theta)/2$ and $\mu' = (v - 2\theta + 2 + \alpha)/4$; substituting $\alpha = -2(\lambda - \mu) - 2$ gives the formula. ∎

Switching by a sign vector is a technique of the papers [12], [30], [34], [40]. It moves a graph between the two lines $\mathcal H(r, s, c_0)$ and $\mathcal H(r, s, \alpha - c_0)$ with the same roots (Section 3). The graph $(36, 16, 14, 8, 6)$ of the papers [12], [13], [25] has a common sign eigenvector of $H$ for $\theta = -8$, and switching gives $(36, 22, 20, 14, 12)$ with $k' = (36 + 8)/2 = 22$ and $\mu' = (36 + 16 - 4)/4 = 12$; its complement $(36, 13, 11, 2, 6)$ is the seed of the graphs of orders 72 and 108 built here (Section 4.8) and of the constructions of the paper [32]. The paper [12] goes the same way from the other end: it switches $(36, 13, 11, 2, 6)$ into $(36, 19, 17, 8, 12)$, the complement of $(36, 16, 14, 8, 6)$; switching commutes with complementation.

### 4.7. What the theorems reach

Table 3 assigns to a theorem each set whose construction a paper names individually, and Table 4 does the same for each family of the papers. Of the individually named sets, Theorem 1 gives 104, Theorem 2 gives 45, Theorem 3 gives 52, Theorem 4 gives 15, Theorem 5 gives 49 and Theorem 6 gives 21; a set with two constructions is counted under both theorems.

The numerical conditions of the theorems also work as a search without graphs. The blow-up of Theorem 4 for $t = \mu$ takes a constructed set to all its multiples, the lift of Corollary 2.2 to the sets $(cu + 1)n$ for every $u$ if the seed has the required partition, Theorem 3(c) to sums of orders on the Hadamard line if the seeds have signed pairs, and condition (7) selects layer gluings. All these conditions are only necessary: the eleven pairs of Section 4.2 satisfy condition (7), yet, by Table 5, there are no graphs. Likewise, condition (7) is passed by $(90, 16, 5, 2, 3)$ on five layers $(18, 4, 3, 0, 1)$ and by $(108, 40, 22, 16, 14)$ on three copies of the graph $(36, 16, 14, 8, 6)$ and on two layers $(36, 16, 14, 8, 6)$ and $(72, 28, 18, 12, 10)$; no paper constructs or excludes these sets. For the first, blocks with constant sums are impossible: $\eta = 3$ requires in-classes of 3 or 1 vertices, while by Lemma 3 a class for the root 1 has size divisible by 6, and a class for the root $-2$ has at least 6 vertices. The gluing of three copies for the second is excluded by Proposition 1; the seeds and quotients of both pairs are in the supplement.

**Proposition 1.** The gluing of Corollary 2.1 of three copies of the graph $(36, 16, 14, 8, 6)$ from the supplement, the graph used in Sections 4.3 and 4.6, does not give $(108, 40, 22, 16, 14)$ for any partitions and orientations.

*Proof.* The graph has roots $4$ and $-2$. With three layers and constant sums, the off-diagonal part of the quotient $Q$ is a matrix of order 3 with zero diagonal and row and column sums $40 - 16 = 24$, that is, with the numbers $a$ and $24 - a$ around the cycle. The diagonal (7) gives $2a(24 - a) = 14\cdot 36 + 2\cdot 16 + 8 - 16^2 = 288$, whence $a = 12$. Hence $\eta = 12 = k - r$ for each pair of layers, and each block has one partition with root $4$ and the other with root $-2$. Let the side of root $4$ be partitioned into $q$ classes, $q \mid 36$. If this is the in-side, then $12 = \ell\cdot 36/q$ and $q = 3\ell$; if it is the out-side, then the row sums of $M$ equal $\ell = q'/3$, and the column sums $q\ell/q' = q/3$ must be integers. In both cases $3 \mid q$, that is, the classes consist of 12, 6, 4, 3, 2 or 1 vertices. By Lemma 3 with $\theta = 4$ the number $a = (16 - 4)w/36 = w/3$ is an integer and $w \ge 5\cdot 36/24 = 7.5$; therefore the sizes 6, 4, 3, 2 and 1 are impossible, and $w = 12$ remains.

A class $F$ of 12 vertices defines a vector $\mathbf 1_F - \tfrac13\mathbf 1$ of the right (for an out-partition) or left (for an in-partition) eigenspace of root $4$. Both spaces are nine-dimensional. In a basis brought to echelon form, a vector of the space is determined by nine free coordinates, and for the vector $\mathbf 1_F - \tfrac13\mathbf 1$ each of them equals $2/3$ or $-1/3$. Therefore a search over all $2^9 = 512$ values of the free coordinates on each side finds all class indicators, and an exact check in rational numbers selects the vectors all of whose coordinates equal $2/3$ or $-1/3$. There is no such vector on either side, right or left. ∎

The same search with the values $\pm\tfrac12$ serves as a control: it finds eight classes of 18 vertices on each side, right and left, so the search does find the objects it looks for. The program and its output are in the supplement.

The proposition does not exclude the pointwise form of Theorem 2 with nonconstant sums, other graphs $(36, 16, 14, 8, 6)$, or a gluing of layers of orders 36 and 72.

### 4.8. Graphs built here

Eight graphs of seven sets are built by the explicit rules of Theorems 2 and 3 from the papers' seeds, and two graphs by a search for pairs by Theorem 1(c). All ten are checked together with their complements by an exact check of equalities (1). The seeds, sign vectors, class labels of the partitions, assignments, matrices and the program that builds each graph by the rule of its theorem and checks it are in the supplement.

1. $(72, 34, 23, 17, 15)$ and $(108, 52, 32, 26, 24)$ are two and three copies of the graph $(36, 16, 14, 8, 6)$ by Theorem 3(c) (the example of Section 4.3). The construction of the first set is claimed in [12], that of the second in [12] and [13]; both are points of the family of the paper [30].
2. $(108, 49, 29, 20, 24)$, the point $m = 9$ of the family $(12m, 6m - 5, 3m + 2, 3m - 7, 3m - 3)$ of the paper [38] on the line $\mathcal H(1, -5, 9)$. The paper assembles it from seeds of orders 36 and 72; here the same set is built in another way, by the gluing of Theorem 3(b) of the graphs $(48, 19, 14, 5, 9)$ and $(60, 25, 17, 8, 12)$ (the second example of Section 4.3).
3. $(66, 31, 17, 14, 15)$, whose construction is claimed in [23] and [46], is the lift of Corollary 2.2 with $c = 2$ of the graph $(22, 9, 6, 3, 4)$, the bordering of the Petersen graph: $r = 1$, $s = -2$, $\theta' = 1$, $c = (9 - 1)/4 = 2$, $m = 3$, $b = 11$. The paper [23] takes the seed from a publication outside the problem, the paper [46] obtains the set by its family at $N = 66$, the paper [25] rechecks the graph of [23], and the paper [44] attributes the set to another paper, although it lies in its own family. An in-partition into two halves of 11 vertices defines a left sign eigenvector for $1$. An exhaustive search over the $2^{11}$ signs on the free coordinates finds 40 such vectors in the 11-dimensional space — 20 partitions into halves, each in the two orders of its classes — and all 40 ordered partitions give lifts $(66, 31, 17, 14, 15)$ that are pairwise distinct as matrices.
4. $(72, 25, 15, 6, 10)$ and $(108, 37, 19, 10, 14)$, whose construction is claimed in [12], are the gluing of Corollary 2.1 of two and three copies of the graph $(36, 13, 11, 2, 6)$ of Section 4.6: $r = 1$, $s = -5$, $\eta = 12 = k - r$, in each block an out-partition for $1$ and an in-partition for $-5$ into three classes of 12 vertices, and $M$ is a permutation matrix; $\mu' = 6 + 144/36 = 10$ and $6 + 2\cdot 4 = 14$. The graph $(72, 25, 15, 6, 10)$ is also constructed independently from a transitive group in the preprint [48].
5. $(144, 52, 26, 20, 18)$ is the lift of Corollary 2.2 with $c = 3$: four copies of the graph $(36, 16, 14, 8, 6)$, an in-partition for $-2$ into three classes of 12 vertices ($c = (16 + 2)/6 = 3$, $b = 12$), with a cyclic and with a noncyclic assignment. The noncyclic assignment tests the freedom of Corollary 2.2 that the π-join [47] lacks: the assignment need not come from a group action.
6. $(75, 30, 14, 9, 14)$ and $(75, 40, 24, 19, 24)$, whose construction is claimed in [16] with data in the supplement to the paper, are given by the converse part of Theorem 1(c). The normals $\alpha = (1, 1, 1, 1, 2, 2, 2)/4$ for $n = 7$ and $\alpha = (1, 1, 1, 1, 1, 1, 3, 3, 3)/8$ for $n = 9$ are taken from the list of hyperplanes of [16]. The pairs $(z_x, \beta_x)$ with multiplicities were found by integer linear programming with the prescribed permutation of coordinates $(0\,1\,2)(4\,5\,6)$ and $(0\,1\,2)(3\,4\,5)(6\,7\,8)$: the first graph has 10 distinct rows $z$, 16 columns $\beta$ and 51 distinct pairs, the second 18, 12 and 57. The exact check of (1) passed for both graphs and their complements $(75, 44, 28, 27, 24)$ and $(75, 34, 18, 17, 14)$. The search method does not affect the result: the matrices and the lists of pairs with multiplicities are in the supplement, and the check does not depend on how they were found.

## 5. Sign-square lines

On the Hadamard line the sign form satisfies $S^2 = \alpha S + \beta I$ (Lemma 2). The simplest case is $\alpha = 0$, when $S$ is a "sign square root" of a scalar matrix. These sets lie on straight lines with step 4 in $v$, and the borderings of the paper [46] work on them.

**Theorem 7.** Let a DSRG lie on the Hadamard line.

1. $S^2 = q^2 I$ for some $q$ if and only if $\mu - \lambda = 1$. Then $q = r - s$ is odd, $r = (q - 1)/2$, $s = -(q + 1)/2$, $t - \mu = (q^2 - 1)/4$, and $S\mathbf 1 = \pm q\mathbf 1$.
2. On such a line $\kappa = (v - q^2 - 1)/4$. Therefore $v \ge q^2 + 5$, and $\kappa = 1$ exactly when $v = q^2 + 5$.
3. A sign matrix of order $n$ with zero diagonal and $S^2 = q^2I$ has $n \ge q^2 + 1$, and equality occurs only for a symmetric one.
4. Suppose the sign form $S$ of the graph has a principal block $S_0$ of order $n < v$ with $S_0^2 = q^2I$, a seed of the same line, and the remaining $m = v - n$ vertices form a principal block $T$; the blocks between $S_0$ and $T$ are arbitrary (bordering of a seed of the same line with side blocks of any rank). Then $n, m \ge q^2 + 1$, and the order of the graph is at least $2q^2 + 4$. If the row and column sums of $S_0$ are constant, then moreover $4 \mid m$ and $m \ge q^2 + 3$.
5. Without the assumption on the line: the sets with $\kappa = 1$ and $c = 1$ whose roots $r \ne s$ are integers and whose multiplicities $f, g$ are integers are exactly

$$
(v, k, t, \lambda, \mu) = (\mu + e + 2m + 1,\ \mu + m,\ \mu + m - 1,\ \mu - 1,\ \mu), \qquad m = \tfrac{q^2 + 3}{4}, \quad e = \tfrac{m^2 + 1}{\mu},
$$

where $q \ge 1$ is odd, $\mu$ is a divisor of $m^2 + 1$ and $q \mid e - \mu$. The complement swaps $\mu$ and $e$. The set lies on the Hadamard line if and only if $\{\mu, e\} = \bigl\{\tfrac{(q - 1)^2 + 4}{4}, \tfrac{(q + 1)^2 + 4}{4}\bigr\}$, that is, $v = q^2 + 5$.

For $q = 1$ the line is part of the series $t = \mu = \lambda + 1$, all feasible sets of which are constructed by the paper [17].

*Proof.* 1. By Lemma 2, $\alpha = -2(r + s + 1)$, and $r + s = \lambda - \mu$; $\alpha = 0$ is equivalent to $\mu - \lambda = 1$. Then $\beta = -(2r + 1)(2s + 1) = q^2$ with $q = 2r + 1 = r - s$, and $t - \mu = -rs = (q^2 - 1)/4$; the row sums are roots of $x^2 - q^2$. 2. The Hadamard line for $\lambda = \mu - 1$ is $v = 4k - 4\mu + 2$, so $\kappa = k - t = \tfrac{v - 2}{4} + \mu - \mu - \tfrac{q^2 - 1}{4} = \tfrac{v - q^2 - 1}{4}$, and $\kappa \ge 1$. 3. $\operatorname{tr} S^2 = \sum_{i \ne j}S_{ij}S_{ji} = q^2 n$, and each term equals $\pm 1$, so $q^2 n \le n(n - 1)$; equality requires $S_{ij}S_{ji} = 1$ for all $i \ne j$, that is, $S = S^{\mathsf T}$. 4. Write $S = \begin{pmatrix} S_0 & U \\ W & T \end{pmatrix}$. The upper left block of $S^2 = q^2I$ is $S_0^2 + UW = q^2I$, whence $UW = 0$. The lower right block is $WU + T^2 = q^2I$, whence $\operatorname{tr}T^2 = q^2m - \operatorname{tr}UW = q^2m$. The principal block $T$ has zero diagonal and signs off it, so the count of item 3 gives $m \ge q^2 + 1$, and for the seed $n \ge q^2 + 1$. Hence $v \ge 2q^2 + 2$; by item 2 $v \equiv 2 \pmod 4$, while $2q^2 + 2 \equiv 0 \pmod 4$, so $v \ge 2q^2 + 4$. If $S_0$ has constant sums $\varepsilon q$, $\varepsilon = \pm 1$, then the matrix $A_0 = (J - I - S_0)/2$ is integral and $4A_0^2 = (n - 2 - 2\varepsilon q)J + (q^2 + 1)I + 2S_0$, so an off-diagonal entry gives $n - 2 - 2\varepsilon q \pm 2 \equiv 0 \pmod 4$, that is, $n \equiv 2\varepsilon q \equiv 2 \pmod 4$. Then $4 \mid m$, and $q^2 + 1 \equiv 2 \pmod 4$ gives $m \ge q^2 + 3$. 5. By (2), $r + s = -c = -1$ and $rs = \mu - t$, so $q = r - s$ is odd, $t - \mu = (q^2 - 1)/4$ and $k = t + 1 = \mu + m$. The equality $(k - r)(k - s) = \mu v$ takes the form $k^2 + k + \mu - t = \mu v$, that is, $\mu(v - 1) = k^2 + 1 = \mu^2 + 2\mu m + m^2 + 1$; hence $\mu \mid m^2 + 1$ and $v - 1 = \mu + e + 2m$. Multiplicities: $f + g = v - 1$ and $k + fr + gs = \operatorname{tr}A = 0$, and with $r = (q - 1)/2$, $s = -(q + 1)/2$ the second equality gives $q(f - g) = v - 1 - 2k = e - \mu$. Hence $f - g$ is an integer if and only if $q \mid e - \mu$, and then $f$ and $g$ are integers: $f + g = \mu + e + 2m \equiv e - \mu \equiv (e - \mu)/q \pmod 2$ for odd $q$. Conversely, the set of the formula satisfies (2) with $r = (q - 1)/2$, $s = -(q + 1)/2$. By the formula of Section 3 the complement has $\bar k = v - k - 1 = e + m$, $\bar t = \bar k - 1$, $\bar\mu - \bar\lambda = 2 - c = 1$ and $\bar\mu = v - 2k + \lambda = e$. Finally, $4k - 2\lambda - 2\mu = 4m + 2 = q^2 + 5$, so the set lies on the Hadamard line exactly when $\mu + e = 2m + 1$. Together with $\mu e = m^2 + 1$ this means that $\mu$ and $e$ are the roots of $x^2 - (2m + 1)x + m^2 + 1$ with discriminant $4m - 3 = q^2$, that is, $\bigl((q \mp 1)^2 + 4\bigr)/4$. ∎

For $q = 5$ the paper [46] obtains $m \ge 26$ by the same trace count for arbitrary side blocks; together with $n \ge 26$ and $v \equiv 2 \pmod 4$ this already gives $v \ge 54$ for any rank. The paper [46] derives the divisibility $4 \mid m$ from the balanced orthogonal sign vectors of rank-one bordering. Item 4 carries the count over to every $q$ and derives the divisibility for any rank from the congruence of orders modulo 4, whence the general bound $m \ge q^2 + 3$. The modules of the paper [46] of orders 12, 28 and 52 for $q = 3, 5, 7$ are minimal: $12 = 3^2 + 3$, $28 = 5^2 + 3$, $52 = 7^2 + 3$. Item 4 also covers the rank-three bordering of the paper [49] (version 3 of the paper [46]): there a module of order 56 is attached to a seed of order 50 for $q = 7$, and the lower left block has rank three.

Items 3 and 4 do not use $\alpha = 0$ in an essential way and carry over to any Hadamard line.

**Corollary 7.1 (bordering on any Hadamard line).** Let $\alpha$, $\beta$ be the numbers of Lemma 2 of a Hadamard line.

(a) A sign matrix of order $n$ with zero diagonal and $S^2 = \alpha S + \beta I$ has $n \ge \beta + 1$, and equality occurs only for a symmetric one.

(b) If such a matrix has $n \ge 2$ and constant row and column sums equal to $c$, then $n \equiv 2c - \alpha \pmod 4$; the residue is the same for both roots $c$ of the polynomial $x^2 - \alpha x - \beta$.

(c) Suppose the sign form of a graph of this line has a principal block $S_0$ of order $n < v$ with $S_0^2 = \alpha S_0 + \beta I$, and the remaining $m = v - n$ vertices form a principal block $T$; the blocks between them are arbitrary. Then $n, m \ge \beta + 1$, and the order of the graph is at least $(\beta + 1) + 4\lceil(\beta + 1)/4\rceil$. If the row and column sums of $S_0$ are constant, then moreover $4 \mid m$ and $m \ge 4\lceil(\beta + 1)/4\rceil$.

*Proof.* (a) The zero diagonal gives $\operatorname{tr}S = 0$, so $\sum_{i \ne j}S_{ij}S_{ji} = \operatorname{tr}S^2 = \beta n$; there are $n(n - 1)$ terms, each equal to $\pm 1$, and the equality $n = \beta + 1$ requires $S = S^{\mathsf T}$. (b) The sums give $c^2 = \alpha c + \beta$. The matrix $A = (J - I - S)/2$ is integral, and $SJ = JS = cJ$ gives $4A^2 = (n - 2 - 2c)J + (\beta + 1)I + (\alpha + 2)S$. The number $\alpha = -2(r + s + 1)$ is even, so $(\alpha + 2)S_{ij} \equiv \alpha + 2 \pmod 4$ for both signs of $S_{ij}$, and an off-diagonal entry gives $n \equiv 2c - \alpha \pmod 4$. The second root $\alpha - c$ gives the residue $\alpha - 2c$, which is congruent to $2c - \alpha$ because this number is even. (c) As in item 4 of Theorem 7, the upper left block of $S^2 = \alpha S + \beta I$ gives $UW = 0$, and the lower right block gives $\operatorname{tr}T^2 = \alpha\operatorname{tr}T + \beta m - \operatorname{tr}UW = \beta m$. The count of part (a) for the principal block $T$ with zero diagonal gives $m \ge \beta + 1$, and for the seed $n \ge \beta + 1$, so $v \ge 2(\beta + 1)$. The sums $c$ of the sign form of a graph of the line are constant, so by (b) $v \equiv 2c - \alpha \pmod 4$. The number $c$ is odd: $c^2 = \alpha c + \beta$, where $\alpha$ is even and $\beta = -(2r + 1)(2s + 1)$ is odd. Hence $(\beta + 1) - (2c - \alpha) = (c - 1)(c - 1 - \alpha)$ is divisible by 4, $v \equiv \beta + 1 \pmod 4$, and the least such $v \ge 2(\beta + 1)$ equals $(\beta + 1) + 4\lceil(\beta + 1)/4\rceil$. If $S_0$ has constant sums, then by (b) $n \equiv v \pmod 4$, so $4 \mid m$, and together with $m \ge \beta + 1$ this is $m \ge 4\lceil(\beta + 1)/4\rceil$. ∎

For $\alpha = 0$ and $\beta = q^2$ these are items 3 and 4: (b) gives $n \equiv 2 \pmod 4$, $q^2 + 1 \equiv 2 \pmod 4$, and $(q^2 + 1) + 4\lceil(q^2 + 1)/4\rceil = 2q^2 + 4$. The bound in (c) needs neither sign vectors of the module, nor rank one of the side blocks, nor constant sums of the seed. Hence it holds for the bordering of Theorem 3 with Lemma 4, for the rank-three bordering of the paper [49] and for any other attachment of vertices to a seed of the same line.

By item 2 the first point of each line is a set with $\kappa = 1$ and $c = 1$, and item 5 reduces all such sets to divisors of the number $m^2 + 1$. For $q = 1, 3, 5, 7, 9$ this is $m^2 + 1 = 2, 10, 50, 170, 442$. Off the Hadamard lines, for $v \le 110$ one pair remains, $(18, 4, 3, 0, 1)$ with the complement $(18, 13, 12, 9, 10)$, where $q = 3$ and $\{\mu, e\} = \{1, 10\}$; the other divisors either give $v > 110$ or violate $q \mid e - \mu$. On the lines these are the points $v = 6, 14, 30, 54, 86$. Further off the lines lie, for example, the pairs with $v = 270, 450, 486$. Theorem 13 of Section 6 applies to all these sets.

Table 2 lists the lines $q = 3, 5, 7, 9$ up to $v = 110$: the first point, the number of points, the orders of the points whose construction the papers under review claim, the points below the bound $2q^2 + 4$ of item 4 and the bound itself.

**Table 2. Sign-square lines for v ≤ 110.**

| $q$ | First point ($\kappa = 1$) | Pairs with $v \le 110$ | Constructed by the papers ($v$) | Below $2q^2 + 4$ ($v$) | $2q^2 + 4$ |
|---|---|---|---|---|---|
| 3 | 14, does not exist | 25 | 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110 | 14, 18 | 22 |
| 5 | 30, does not exist | 21 | 34, 50, 54, 58, 62, 66, 70, 74, 82, 86, 90, 94, 98, 102, 106, 110 | 30, 34, 38, 42, 46, 50 | 54 |
| 7 | 54, constructed | 15 | 54, 82, 98, 102, 106 | 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98 | 102 |
| 9 | 86, not settled by the papers | 7 | none | 86, 90, 94, 98, 102, 106, 110 | 166 |

The points of a line with $v < 2q^2 + 4$ are not constructed by bordering of a seed of the same line for any rank of the side blocks: the sign form of such a graph has no proper principal block with $S_0^2 = q^2I$. They need another construction. For $q = 5$ these are the points $v = 30, \ldots, 50$, and for $q = 7$ the points $v = 54, \ldots, 98$; on the line $q = 9$ the bound 166 lies above all points with $v \le 110$.

On the lines $\mathcal H(4, -2, 3)$ and $\mathcal H(1, -5, 9)$, $\alpha = \mp 6$ and $\beta = 27$, and by Corollary 7.1 bordering of a seed of the same line gives on them orders at least $28 + 28 = 56$ for any rank of the blocks. Hence it does not construct the points $(44, 20, 16, 10, 8)$ and $(52, 24, 18, 12, 10)$ of the first line or $(40, 15, 12, 3, 7)$, $(44, 17, 13, 4, 8)$ and $(52, 21, 15, 6, 10)$ of the second; none of the papers under review settles these pairs (Section 9). The point 68 of Section 9.6 lies above the bound.

## 6. Obstructions

### 6.1. The denominator

**Theorem 8.** Let $M$ be a zero-one matrix of order $v$ and rank $n$ with constant row and column sums $K$, and let $v/K = p/q$ in lowest terms. Then for every invertible submatrix $N_0$ of order $n$ lying on $n$ independent rows and $n$ independent columns, the number $h = |\det N_0|/q$ is an integer and

$$
h \le \frac{D(n)}{q}, \qquad h\,|2q - p| \le D(n - 1). \tag{10}
$$

In particular, $q \le D(n)$, and $|2q - p| \le D(n - 1)$ when $p \ne 2q$. For a DSRG with $t = \mu$ the theorem is applied to $M = A$ ($K = k$) and to $M = J - A$ ($K = v - k$) with $n = k/d + 1$. If (10) fails on at least one side, there is no graph — and there are no graphs for any multiple of the parameters, $(vm, km, \mu m, \lambda m, \mu m)$, because $n$ and $v/K$ do not depend on $m$.

*Proof.* Take $n$ independent columns $C$ of $M$ and an invertible submatrix $N_0$ on $n$ rows of $C$. The vector $M\mathbf 1 = K\mathbf 1$ lies in the column space, so $C\alpha = \mathbf 1$; on the chosen rows $N_0\alpha = \mathbf 1$, and, as in Theorem 1(c), $\mathbf 1^{\mathsf T}\alpha = v/K$. By Cramer's rule $\alpha_j = \det N_j/\det N_0$, where $N_j$ is the matrix $N_0$ with its $j$-th column replaced by ones, again a zero-one matrix. Therefore $(v/K)\det N_0 = \sum_j \det N_j \in \mathbb Z$, and $q \mid \det N_0$. Next, $J - 2N_0 = -2N_0 + \mathbf 1\mathbf 1^{\mathsf T}$, and by the matrix determinant lemma for a rank-one perturbation $\det(J - 2N_0) = (-2)^n\det N_0\,(1 - \mathbf 1^{\mathsf T}N_0^{-1}\mathbf 1/2) = (-2)^n\det N_0\,(1 - \tfrac{p}{2q})$. The matrix $J - 2N_0$ consists of $\pm 1$; multiplying its rows and columns by $\pm 1$ so that the first row and column become all ones, and subtracting the first row from the others, we get $|\det(J - 2N_0)| = 2^{n - 1}|\det B|$ with a zero-one $B$ of order $n - 1$. Hence $|\det N_0|\cdot|2q - p|/q = |\det B| \le D(n - 1)$, while $|\det N_0| \le D(n)$. ∎

The exact values are $D(1), \ldots, D(13) = 1, 1, 2, 3, 5, 9, 32, 56, 144, 320, 1458, 3645, 9477$ [8]; the Hadamard bound $D(m) \le (m + 1)^{(m + 1)/2}/2^m$ also serves.

**What Theorem 8 closes.** $(54, 15, 5, 2, 5)$: $d = 3$, $n = 6$, on the side of $A$ $v/k = 18/5$, $|2q - p| = 8 > D(5) = 5$. $(96, 20, 5, 1, 5)$: $n = 6$, $v/k = 24/5$, $|2q - p| = 14 > 5$. Hence the nonexistence of the rays $(54m, 15m, 5m, 2m, 5m)$ and $(96m, 20m, 5m, m, 5m)$ for all $m$ and of their complements; for $v \le 110$ these are $(54, 15, 5, 2, 5)$, $(96, 20, 5, 1, 5)$ and $(108, 30, 10, 4, 10)$. The papers [13] and [33] excluded these sets by an exhaustive enumeration of hyperplanes; Theorem 8 gives a one-line argument. For $n \ge 12$ the theorem excludes nothing at all: $D(n - 1) \ge D(11) = 1458$ exceeds $2v$.

**Where else it works.** Theorem 8 bounds the search over normals. For the ray $(72, 21, 7, 4, 7)$ of the paper [15]: $n = 8$, $v/k = 24/7$, $|\det N_0| \le D(7)\cdot 7/10 = 22.4$ with $7 \mid \det N_0$, that is, $\det N_0 \in \{7, 14, 21\}$ and $h \le 3$. For the pair $(81, 48, 32, 26, 32) \leftrightarrow (81, 32, 16, 15, 11)$, about which the papers under review claim nothing: $d = 6$, $n = 9$; on the side of $A$ $v/k = 27/16$ and $h \le \min(144/16, 56/5) = 9$, on the side of $J - A$ $v/(v - k) = 27/11$ and $h \le \min(144/11, 56/5)$, that is, $h \le 11$.

### 6.2. The cone obstruction

**Theorem 9.** Let $t = \mu$, $g = k/d$, $n = g + 1$. Divide the identity (4) by $d$ and define the weight of a pair as $w(z, \beta) = |\{x : (z_x, \beta_x) = (z, \beta)\}|/d \ge 0$. Then

$$
\sum_{(z, \beta)} w(z, \beta)\,\beta z^{\mathsf T} = \frac{n}{v/k}\,\alpha\mathbf 1^{\mathsf T} - I, \qquad \mathbf 1^{\mathsf T}\alpha = v/k, \tag{11}
$$

where the pairs are subject to the conditions $z \in \{0, 1\}^n$, $z^{\mathsf T}\alpha = 1$, $\mathbf 1^{\mathsf T}\beta = 1$, $z^{\mathsf T}\beta = 0$ and $z'^{\mathsf T}\beta \in \{0, 1\}$ for all $z'$ that occur. The system (11) depends only on $g$ and $v/k$. Therefore its unsolvability for all $\alpha$ excludes all sets $(v, gd, \mu, \mu - d, \mu)$ with these $g$ and $v/k$, including all multiples $(vm, km, \mu m, \lambda m, \mu m)$. Conversely, a single constructed graph makes the ratio $v/k$ admissible for all such sets.

*Proof.* $\mu/d = nk/v$, because $k(k + d) = \mu v$ and $k + d = dn$. Dividing (4) by $d$ gives (11), and the conditions on the pairs are Theorem 1(c). ∎

Theorem 9 explains why the papers [13], [15], [19], [21], [33] close whole rays: the catalogues of normals $\alpha$ and the certificates do not depend on $m$. A nonexistence certificate for a fixed collection of pairs is a matrix $Y$ with $\langle Y, \beta z^{\mathsf T}\rangle = \beta^{\mathsf T}Yz \ge 0$ on all admissible pairs and $\langle Y, \tfrac{nk}{v}\alpha\mathbf 1^{\mathsf T} - I\rangle < 0$, where $\langle Y, M\rangle = \operatorname{tr}(Y^{\mathsf T}M) = \sum_{i, j}Y_{ij}M_{ij}$ (Farkas' lemma). The special case $Y = e_je_j^{\mathsf T}$: a coordinate with $\alpha_j < d/\mu$ requires a negative $\beta_x(j)$. The ratios $v/k$ with $g = 5$ that the papers [13] and [33] exclude are excluded here anew: two by Theorem 8, and the rest by a search with integer Farkas certificates (Section 6.7, item 5).

### 6.3. Twins and the quotient by row types

**Theorem 10.** (a) In every DSRG a class of vertices with one and the same in-neighbourhood contains at most $v - 2k + \mu - 1$ vertices; the same holds for out-neighbourhoods.

(b) Let $t = \mu$, let the rows of $A$ take $T$ distinct values with classes of sizes $w_1, \ldots, w_T$, and let $Q_{ij}$ be the number of out-neighbours of a vertex of class $i$ in class $j$. Then $Q^2 + dQ = \mu\mathbf 1w^{\mathsf T}$, the diagonal of $Q$ is zero, the spectrum of $Q$ is $k$, $(-d)^g$ and $0^{T - n}$, and $T \ge n$. When $T = n$ all classes have size $dk/\mu$, and every vertex receives arcs from exactly $\mu/d$ classes. The matrix $E = Q + dI - \tfrac{k + d}{v}\mathbf 1w^{\mathsf T}$ has rank $T - n$.

*Proof.* (a) Let $S_0$ be the common in-neighbourhood of a class $U_0$. $U_0 \cap S_0 = \varnothing$, since otherwise there is a loop, and there are no arcs inside $U_0$. For $w \notin S_0$ and $y \in U_0$, $y \ne w$, there is no arc $w \to y$, so $|N^+(w) \cap S_0| = (A^2)_{wy} = \mu$. The set $V \setminus (S_0 \cup N^+(w))$ of $v - 2k + \mu$ vertices contains $U_0$ and also $w$ itself if $w \notin U_0$. Such a $w \notin S_0 \cup U_0$ exists, because $|S_0 \cup U_0| \le k + (v - 2k + \mu) < v$: in every DSRG $\mu \le t < k$ (a necessary condition of Duval [1]).

(b) When $t = \mu$ the equality (1) takes the form $A^2 + dA = \mu J$. Let $U_1$ be the $v \times T$ matrix of class indicators and $V_1$ the $T \times v$ matrix of distinct rows; then $A = U_1V_1$ and $Q = V_1U_1$. For a vertex $x$ of class $i$ the number $Q_{ij}$ equals $\sum_{z \in j}A_{xz}$, and $(Q^2)_{ij}$ equals the number of paths of length 2 from $x$ into class $j$, that is, $\sum_{z \in j}(A^2)_{xz}$. Therefore $(Q^2 + dQ)_{ij} = \sum_{z \in j}(A^2 + dA)_{xz} = \mu w_j$. The diagonal of $Q$ is zero: an arc $x \to y$ inside a class would give a loop at $y$, which has the same row. The nonzero eigenvalues of $U_1V_1$ and $V_1U_1$ coincide with multiplicities, and for $A$ these are $k$ and $(-d)^g$, because $r = 0$, $s = -d$ and $\operatorname{rank}A = n$; this gives the spectrum of $Q$ and $T \ge n$. The row sums of $Q$ equal $k$, and $w^{\mathsf T}Q = kw^{\mathsf T}$, because $kw_j$ arcs enter class $j$. Therefore $w^\perp$ is invariant, $Q(Q + dI) = 0$ on it, and $Q$ is diagonalizable on $w^\perp$ with eigenvalues $0$ and $-d$. When $T = n$ the matrix $Q$ is invertible, and from $Q\mathbf 1 = k\mathbf 1$ we get $Q + dI = \mu Q^{-1}\mathbf 1w^{\mathsf T} = (\mu/k)\mathbf 1w^{\mathsf T}$; the diagonal gives $w_i = dk/\mu$. The in-neighbourhood of a vertex is a union of classes, and the in-degree $k$ equals $dk/\mu$ times the number of these classes. Finally, $E\mathbf 1 = 0$, and on $w^\perp$ the matrix $E$ coincides with $Q + dI$, whose rank there equals the dimension of the kernel of $Q$, that is, $T - n$. The space is the sum of $\langle\mathbf 1\rangle$ and $w^\perp$, so $\operatorname{rank}E = T - n$. ∎

**What it closes.** For two rays Theorem 10 closes only the cases with a small number of row types; the remaining cases, and with them the whole rays, are closed by the computations of Section 6.7 (items 2 and 3). The ray $(36m, 21m, 14m, 11m, 14m)$ of the paper [21]: $d = 3m$, and when $T = n$ one needs $\mu/d = 14/3$ source classes, which is not an integer for any $m$. The ray $(28m, 21m, 18m, 15m, 18m)$ of the paper [19] for odd $m$: $n = 8$, $(k + d)/v = 6/7$, and $7E = 7Q + 21mI - 6\cdot\mathbf 1w^{\mathsf T}$ is an integer matrix with $7E\mathbf 1 = 0$ and odd diagonal $21m - 6w_i$. When $T = n$ the class size $dk/\mu = 7m/2$ is not an integer. When $T = n + 1 = 9$ the rank of $7E$ equals 1, and modulo 2 the rank is at most 1; a matrix of rank 1 over $\mathbb F_2$ with unit diagonal is $uv^{\mathsf T}$, and $u_iv_i = 1$ gives $u = v = \mathbf 1$, that is, $J$ of order 9. Then the row sum is odd, whereas $7E\mathbf 1 = 0$. Hence $T \notin \{n, n + 1\}$. The paper [19] excludes the points $m = 1$ and $m = 3$ by direct proofs with computations, and all odd $m$ by a general proof in which the scale does not enter (Section 6.7, item 2). The twin bound (a) contains as a special case the lemma of the paper [19] on classes of at most $4m - 1$ vertices.

### 6.4. The rank of the skew-symmetric part

Call a vertex set $X$ *N-balanced* if $D\mathbf 1_X = 0$: every vertex has as many one-way out-arcs into $X$ as one-way in-arcs from $X$. Call a union of weak components of $N$ *N-closed*; it is N-balanced. For an N-closed $C$ the matrix $D$ is block diagonal, $D = D_C \oplus D_{\bar C}$, and $A_C = A[C, C]$.

**Theorem 11.** (a) The image of $D$ lies in $\mathbf 1^\perp$ and is invariant under $A$ and $A^{\mathsf T}$; its dimension is $2m'$, and $A$ has on it the eigenvalues $r$ and $s$, exactly $m'$ times each. Therefore $\operatorname{rank} D = 2m' \le 2\min(f, g)$. The kernel of $D$ is invariant, and $A = A^{\mathsf T}$ on it.

(b) For $H = A + A^{\mathsf T} + cI$ we have $HD + DH = 0$ and $H^2 = (r - s)^2I + DD^{\mathsf T} + 4\mu J$.

(c) Let $f \ne g$, let $\theta$ be the eigenvalue of larger multiplicity and $\operatorname{rank} D = 2\min(f, g)$ (*the full-rank case*; a smaller rank is *the rank-deficient case*). Then $\ker D \cap \mathbf 1^\perp$ has dimension $|f - g|$, and $A = A^{\mathsf T} = \theta I$ on it. Every N-balanced $X \ne V$ satisfies $A\mathbf 1_X = \theta\mathbf 1_X + \beta_X\mathbf 1$ with an integer $\beta_X = (k - \theta)|X|/v$.

(d) In the full-rank case, for an N-closed $C \ne V$ of $n_C$ vertices, $\dim\ker D_C = n_C\,\delta/v$, where $\delta = \dim\ker D = |f - g| + 1$. In particular, $v \mid n_C\delta$.

(e) Let $M = v(A - rI) - (k - r)J$ (rank $g$) and let $B_K$ be the principal submatrix of $M$ on a vertex set $K$ with $\operatorname{rank} B_K = \operatorname{rank} M$. Then the row $L_i = M[i, K]$ and the column $R_i = M[K, i]$ of a vertex $i \notin K$ lie in the row and column spaces of $B_K$, and $M_{ij} = L_iGR_j$ for all $i, j \notin K$ and every $G$ with $B_KGB_K = B_K$. Two vertices outside $K$ with equal $L$ and $R$ are impossible if the diagonal value of $M$ does not occur off the diagonal. The same holds with $s$ in place of $r$ (rank $f$).

*Proof.* (b) By (1) and (2), $(A + \tfrac c2 I)^2 = \tfrac{(r - s)^2}{4}I + \mu J$, and the same for $A^{\mathsf T}$. Since $2A + cI = H + D$ and $2A^{\mathsf T} + cI = H - D$, we get $(H + D)^2 = (H - D)^2 = (r - s)^2I + 4\mu J$. The difference of the squares equals $2(HD + DH) = 0$, the sum equals $2H^2 + 2D^2$, and $D^2 = -DD^{\mathsf T}$.

(a) From $HD = -DH$ it follows that $D^2$ commutes with $H$ and $D$, and hence with $A$ and $A^{\mathsf T}$. The matrix $D^2$ is symmetric, and its eigenspaces are invariant. The kernel of $D^2$ coincides with the kernel of $D$ and contains $\mathbf 1$. On the subspace $W_\sigma$ where $D^2 = -\sigma^2 I$, $\sigma > 0$, the matrix $J$ is zero and $Y_\sigma = (2A + cI)|_{W_\sigma}$ satisfies $Y_\sigma^2 = (r - s)^2I$. The restriction of $D$ to $W_\sigma$ is invertible and skew-symmetric, so $\operatorname{tr} D|_{W_\sigma} = 0$, and $H|_{W_\sigma} = -D H D^{-1}|_{W_\sigma}$ gives $\operatorname{tr} H|_{W_\sigma} = 0$. Hence $\operatorname{tr} Y_\sigma = 0$, and the values $\pm(r - s)$ occur equally often: on $W_\sigma$ the eigenvalues $r$ and $s$ of $A$ occur equally often. The image of $D$ is the sum of all $W_\sigma$. On $\ker D$ we have $Ax - A^{\mathsf T}x = Dx = 0$.

(c) $A$ is diagonalizable, and the space decomposes into the sum of the invariant subspaces $\operatorname{im} D$, $\ker D \cap \mathbf 1^\perp$ and $\langle\mathbf 1\rangle$; therefore on $\ker D \cap \mathbf 1^\perp$ the eigenvalues $r$ and $s$ have multiplicities $f - m'$ and $g - m'$, and in the full-rank case the smaller of them is zero. For an N-balanced $X$ the vector $\mathbf 1_X - \tfrac{|X|}{v}\mathbf 1$ lies in $\ker D \cap \mathbf 1^\perp$, and Lemma 3 gives the rest.

(d) The block $(C, C)$ of the equality $HD + DH = 0$ gives $H_CD_C + D_CH_C = 0$ for $H_C = H[C, C]$. Hence $\ker D_C$ and $\operatorname{im} D_C$ are invariant under $H_C$ and $D_C$, and thus under $2A_C + cI = H_C + D_C$. In an orthonormal basis of $\operatorname{im} D_C$ the restrictions are a symmetric $H'$, an invertible skew-symmetric $D'$ and $Y = H' + D'$; from $D'H' = -H'D'$ it follows that $D'YD'^{-1} = -H' + D' = -Y^{\mathsf T}$, so the spectrum of $Y$ is symmetric about zero and $\operatorname{tr} Y = 0$. The diagonal of $A_C$ is zero, so the trace of $A_C$ on $\ker D_C$ equals $c(n_C - d_C)/2$, where $d_C = \dim\ker D_C$. In the full-rank case a vector $x \in \ker D_C$ orthogonal to $\mathbf 1_C$ and extended by zero lies in $\ker D \cap \mathbf 1^\perp$, so $A_Cx = \theta x$; and $A_C\mathbf 1_C = (\theta + \beta_C)\mathbf 1_C$ by (c). Comparing traces: $\theta d_C + \beta_C = c(n_C - d_C)/2$, that is, $d_C(2\theta + c) = n_C(cv - 2k + 2\theta)/v$. For $C = V$ the same argument with $\beta = k - \theta$ gives $\delta(2\theta + c) = cv - 2k + 2\theta$. The number $2\theta + c = \pm(r - s)$ is not zero, whence $d_C = n_C\delta/v$. Finally, $\delta = v - 2m' = v - 2\min(f, g) = |f - g| + 1$.

(e) $M\mathbf 1 = 0$, and $M$ annihilates the right eigenvectors for $r$ and multiplies by $v(s - r)$ the right eigenvectors for $s$ orthogonal to $\mathbf 1$; therefore $\operatorname{rank} M = g$. The principal submatrix of $M$ on $K \cup \{i, j\}$ has rank at most $\operatorname{rank} M = \operatorname{rank} B_K$, and this is equivalent to $R_j$ lying in the column space of $B_K$, $L_i$ lying in the row space, and the Schur complement being zero: $M_{ij} = L_iGR_j$. If $L_i = L_j$ and $R_i = R_j$, then $M_{ij} = L_iGR_i = M_{ii}$. ∎

**What Theorem 11 closes.** For $(27, 7, 4, 1, 2)$, $r = 1$, $s = -2$, $f = 15$, $g = 11$. In the full-rank case $\operatorname{rank} D = 22$, $\theta = 1$, $\delta = 5$, and N-balanced sets have $\beta_X = 2|X|/9 \in \mathbb Z$, that is, 9 or 18 vertices. A component of $N$ with 9 or 18 vertices would give $d_C = 5n_C/27 \notin \mathbb Z$. Hence in the full-rank case the one-way part $N$ is connected; the paper [28] proves connectivity in the rank-deficient case as well, by computation. If $N$ is a Cayley graph of a group $G$ with connection set $S_0$, then part (a) requires $\operatorname{rank}(N - N^{\mathsf T}) \le 2\min(f, g)$ for any reciprocal part. The paper [28] excludes Cayley graphs of the five groups of order 27 by solving 207 SAT formulas with DRAT proofs. Part (a) adds to this a filter on the one-way part alone; here it is applied by an exhaustive search, and the program and its output are in the supplement. The one-way part of a Cayley graph is the Cayley graph of a set $S_0$ of $\kappa = 3$ elements containing neither the identity nor a pair of mutually inverse elements; in a group of order 27 there are $\binom{13}{3}\cdot 2^3 = 2288$ such sets, and 11,440 in the five groups. For each of them the rank of $D = N - N^{\mathsf T}$ was computed: first modulo the prime 32,749 — this rank is at most the rational one, and a value of at least 24 excludes the set at once — and for the rest by an exact computation in rational numbers. The rank 22 never occurred: for all sets with rank at most 22 it is at most 20. Hence in the full-rank case a Cayley graph is excluded without SAT formulas, and the search of [28] is needed only in the rank-deficient case. Part (e) is the basis of the proof [43] for $(30, 11, 9, 2, 5)$: there $r = 1$, $s = -4$, $(k - r)/v = 1/3$, and $M/(-10) = 3I - 3A + J$ has rank $g = 8$. The method (e) applies to sets with small $\min(f, g)$, for example to $(40, 15, 12, 3, 7)$ with $g = 9$ and to $(81, 32, 16, 15, 11)$ with $g = 8$, about which the papers under review claim nothing.

### 6.5. Arc defects and reciprocal triangles

**Lemma 5 (identity of the one-way part).** $UD + DU + N^2 - (N^{\mathsf T})^2 = (\lambda - \mu)D$.

*Proof.* The difference of (1) and the transpose of (1) gives $A^2 - (A^{\mathsf T})^2 = (\lambda - \mu)(A - A^{\mathsf T})$. Substituting $A = U + N$, $A^{\mathsf T} = U + N^{\mathsf T}$ expands the left-hand side into $UN + NU + N^2 - UN^{\mathsf T} - N^{\mathsf T}U - (N^{\mathsf T})^2$. ∎

For a one-way arc $e = (x \to y)$ denote by $N^\pm(\cdot)$ the one-way out- and in-neighbours and put: $\iota(e) = |N^+(y) \cap N^-(x)|$, the number of cyclic triangles of $N$ through $e$; $\tau(e) = |N^+(y) \cap N^+(x)| + |N^-(x) \cap N^-(y)|$, the number of transitive triangles of $N$ in which $e$ is a short side; $\zeta(e)$, the number of $w \in N^+(y)$ not adjacent to $x$ in either direction plus the number of $w \in N^-(x)$ not adjacent to $y$; $\rho(e) = U(x, N^-(y) \setminus x) + U(N^+(x) \setminus y, y) + |(N^+(x) \setminus y) \cap (N^-(y) \setminus x)|$. Let $C_3$, $T_3$ and $Z_3$ be the numbers of cyclic triangles of $N$, transitive triangles of $N$, and paths $x \to y \to w$ in $N$ with nonadjacent ends.

**Theorem 12.** (a) For every one-way arc, $\iota(e) + \tau(e) + \zeta(e) + \rho(e) = \sigma = 2\kappa - c$. Therefore $c \le 2\kappa$ (Duval's inequality); when $\sigma = 0$ there are no triangles in $N$ and the ends of paths of length 2 in $N$ are reciprocal; when $\sigma = 1$ every arc has exactly one defect.

(b) $\operatorname{tr}U^3 + 3(C_3 + T_3 + Z_3) = (\lambda - \mu)vt + \mu vk + 3v\kappa(\kappa - \mu)$, and the number of reciprocal triangles is exactly $\operatorname{tr}U^3/6$.

*Proof.* (a) The entry $(x, y)$ of Lemma 5: $(UD)(x, y) = U(x, N^-(y)) - U(x, N^+(y))$, $(DU)(x, y) = U(N^+(x), y) - U(N^-(x), y)$, $N^2(x, y) = |N^+(x) \cap N^-(y)|$, $(N^{\mathsf T})^2(x, y) = |N^+(y) \cap N^-(x)|$, and $D(x, y) = 1$. We get $\rho(e) = (\lambda - \mu) + U(x, N^+(y)) + U(N^-(x), y) + \iota(e)$. Of the $2\kappa$ terms $U(x, w)$, $w \in N^+(y)$, and $U(w, y)$, $w \in N^-(x)$, exactly those vanish where the pair is not reciprocal: a cyclic triangle (such a $w$ kills two terms), a transitive one, or nonadjacency. Therefore $U(x, N^+(y)) + U(N^-(x), y) = 2\kappa - 2\iota - \tau - \zeta$, and $\iota + \tau + \zeta + \rho = 2\kappa - c$. All defects are nonnegative.

(b) $\operatorname{tr}A^3 = (\lambda - \mu)\operatorname{tr}A^2 + \mu\operatorname{tr}AJ = (\lambda - \mu)vt + \mu vk$. By the cyclicity of the trace, $\operatorname{tr}A^3 = \operatorname{tr}U^3 + 3\operatorname{tr}U^2N + 3\operatorname{tr}UN^2 + \operatorname{tr}N^3$. The entry $(x, y)$ of the equality (1) on a one-way arc gives $\lambda = U^2(x, y) + \rho(e)$, so $\operatorname{tr}U^2N = \lambda v\kappa - \sum_e\rho(e)$. The trace $\operatorname{tr}UN^2$ counts paths of length 2 in $N$ with reciprocal ends: there are $v\kappa^2$ paths in all, of which a transitive triangle gives one path with one-way ends, a cyclic one gives three, and nonadjacent ends give the $Z_3$ paths; therefore $\operatorname{tr}UN^2 = v\kappa^2 - T_3 - 3C_3 - Z_3$ and $\operatorname{tr}N^3 = 3C_3$. Summing (a) over all arcs gives $\sum_e\rho(e) = (2\kappa - c)v\kappa - 3C_3 - 2T_3 - 2Z_3$. Substitution gives the identity. ∎

Part (a) is Lemma 3 of the paper [37] and equality (17) of the paper [43]; equality (18) there is its form for an induced subgraph, where the right-hand side is at most $\sigma$. **What it closes.** The line $(6d, 2d + 1, 2d - 1, 2, d)$ of the paper [43] for $d \ge 7$: $\kappa = 2$, while $c = d - 2 > 2\kappa$. Together with Theorem 11(e) and a computation, it closes the set $(30, 11, 9, 2, 5)$, which has $\sigma = 1$.

### 6.6. The one-way part when $\kappa = 1$

When $\kappa = 1$ the matrix $N$ is the matrix of a permutation $\pi$: a single unit entry in each row and each column.

**Theorem 13.** Let $\kappa = 1$.

1. $\pi$ has no fixed points and no cycles of length 2, and $c \in \{0, 1, 2\}$. For $a(x) = U(x, \pi^2x)$ we have $a(x) + a(\pi^{-1}x) + [\pi^3x = x] = c$. Therefore cycles of length 3 occur only when $c = 1$, and when $c \in \{0, 2\}$ all cycles have length at least 4.
2. *Alternation.* Let $c = 1$ and $u_\ell(x) = U(x, \pi^\ell x)$ for $x$ on a cycle of length $L \ge 4$. Then for every $2 \le \ell \le L - 2$ the function $u_\ell$ alternates along the cycle: $u_\ell(\pi x) = 1 - u_\ell(x)$. Therefore the cycle lengths are 3 or multiples of 4, and the number $a_3$ of cycles of length 3 satisfies $3a_3 \equiv v \pmod 4$.
3. *Fourier window.* Let $c = 1$, a cycle of length $L \ge 4$, $\omega = e^{i\vartheta}$, $\vartheta = 2\pi j/L$ with $\sin\vartheta \ne 0$, and $\varepsilon_\ell = 2u_\ell(x_0) - 1$ for a fixed vertex $x_0$ of the cycle. Then
$$
\Bigl|\sum_{\ell = 2}^{L - 2}\varepsilon_\ell\,\omega^\ell\Bigr|^2 \le 4(t - \mu) + 5 - 4\cos^2\vartheta, \qquad \varepsilon_{L - \ell} = (-1)^\ell\varepsilon_\ell,
$$
with equality if no other cycle carries a frequency with sine $-\sin\vartheta$. For a cycle of length 3, the other cycles of length divisible by 3 contain at least $3(t - \mu + 1)$ vertices.
4. *Edge count.* When $c = 1$, the vertices of a cycle of length $L \ge 4$ have in total $L(2t + 3 - L)/2$ reciprocal neighbours outside the cycle, so $L \le 2t + 3$. If there are exactly $a$ cycles of length 3, then $\sum L(2t + 3 - L)/2 \ge 3a\max\bigl(0,\ t - 3(a - 1)\bigr)$, where the sum runs over the cycles of length at least 4.
5. *Quotient by the orbits of $\pi^2$.* $\ker D$ consists of the functions constant on the orbits of $\pi^2$. The partition into these orbits is equitable for $U$ with quotient $B_U$, $\pi$ permutes the orbits by a permutation $\Pi$, and with the orbit sizes $w$ we have $(2B_U + 2\Pi + cI)^2 = (r - s)^2I + 4\mu\mathbf 1w^{\mathsf T}$.
6. *Triples only.* If $c = 1$ and all cycles have length 3, then the block of $U$ between two triples is $0$, $J$, the matrix $Q_0$ of one of the three transpositions of the triple, or $J - Q_0$, and the quotient $C_U$ (the block counts) satisfies $C_U^2 + 3C_U - \tfrac{(r - s)^2 - 9}{4}I = 3\mu J$.

*Proof.* 1. There is no fixed point since there are no loops; a 2-cycle $x \to y \to x$ would make the pair reciprocal. The entry $(x, \pi x)$ of Lemma 5: $(UD)(x, \pi x) = -a(x)$, $(DU)(x, \pi x) = -a(\pi^{-1}x)$, $N^2(x, \pi x) = 0$, $(N^{\mathsf T})^2(x, \pi x) = [\pi^3x = x]$, and the sum equals $\lambda - \mu$. On a cycle of length 3, $\pi^2x = \pi^{-1}x$, and the arc $\pi^{-1}x \to x$ is one-way, so $a = 0$ and $c = 1$. The inequality $0 \le c \le 2$ is Theorem 12(a), together with $c \ge 0$ from the equality on a cycle of length at least 4 ($a \ge 0$).

2. The entry $(x, \pi^jx)$ of Lemma 5 for $1 \le j \le L - 3$ gives $u_{j - 1}(x) + u_{j - 1}(\pi x) + [j \le 2] = u_{j + 1}(x) + u_{j + 1}(\pi^{-1}x)$, where $u_0 = u_1 = 0$. By induction on $j$ the left-hand side equals one, and $u_{j + 1}$ alternates for $j + 1 \le L - 2$. For $L = 2m_0$ with odd $m_0 \ge 3$, the symmetry of $U$ gives $u_{m_0}(\pi^{m_0}x) = u_{m_0}(x)$, while alternation over an odd number of steps gives the opposite value; odd $L \ge 5$ are excluded by alternation at $\ell = 2$. Thus $L \in \{3\} \cup 4\mathbb Z$, and $v \equiv 3a_3 \pmod 4$.

3. The vectors $e(\pi^ix_0) = \omega^i/\sqrt L$ on the cycle and $0$ off it are eigenvectors of $N$ with eigenvalue $\omega$ and of $D$ with eigenvalue $2i\sin\vartheta$. Substitute them into Lemma 5: for a vector $f \ne e$ of the same basis with frequency $\vartheta'$ the terms $N^2 - (N^{\mathsf T})^2$ and $(\lambda - \mu)D$ give zero, and we get $2i(\sin\vartheta + \sin\vartheta')\langle f, Ue\rangle = 0$; for $f = e$ they do not cancel, and the same inner product gives $\langle e, Ue\rangle = -(1 + 2\cos\vartheta)/2$. Hence $Ue$ is the component $\langle e, Ue\rangle e$ plus a vector in the frequencies with sine $-\sin\vartheta$. The sum of (1) and the transpose of (1), in the inner product with $e$, gives $\|Ue\|^2 = (t - \mu) + 3/2 + \cos\vartheta$. By item 2, $u_\ell(\pi^ix_0) = \tfrac12 + \tfrac{(-1)^i}{2}\varepsilon_\ell$ for $2 \le \ell \le L - 2$, and $u_0 = u_1 = u_{L - 1} = 0$; since $L$ is even, the contribution of the cycle itself to $\|Ue\|^2$ equals $\bigl|\tfrac12\sum_{\ell = 2}^{L - 2}\omega^\ell\bigr|^2 + \bigl|\tfrac12\sum_{\ell = 2}^{L - 2}\varepsilon_\ell\omega^\ell\bigr|^2 = \tfrac{(1 + 2\cos\vartheta)^2}{4} + \tfrac14\bigl|\sum\varepsilon_\ell\omega^\ell\bigr|^2$. It is at most $\|Ue\|^2$ and equals it if there are no other partners; this gives the inequality. The relation $\varepsilon_{L - \ell} = (-1)^\ell\varepsilon_\ell$ is the symmetry of $U$. For a triple and $\vartheta = 2\pi/3$ the vector $Ue$ lies on the cycles of length divisible by 3; each of its coordinates is a sum of zero to three distinct cube roots of unity divided by $\sqrt3$, so its squared modulus is at most $1/3$, while $\|Ue\|^2 = t - \mu + 1$.

4. By item 2, for $2 \le \ell \le L - 2$ the function $u_\ell$ equals one at exactly $L/2$ vertices, so inside the cycle the sum of $U$ over ordered pairs equals $(L - 3)L/2$, and $Lt - (L - 3)L/2$ remains outside the cycle. A vertex of a triple has no reciprocal neighbours in its own triple — its pairs are joined by arcs of $\pi$ — and has at most $3(a - 1)$ of them on the other triples; hence at least $t - 3(a - 1)$ of its reciprocal neighbours lie on the long cycles. By the symmetry of $U$ the sum of these numbers over the $3a$ vertices of the triples is at most the number of reciprocal neighbours of the long cycles outside their own cycles.

5. With $Nx(y) = x(\pi y)$ the condition $Dx = 0$ reads $x(\pi y) = x(\pi^{-1}y)$. By Theorem 11(a), $\ker D$ is invariant under $A$, $A^{\mathsf T}$ and, by (b), under $H$, and $DD^{\mathsf T} = 0$ on it; hence $H^2 = (r - s)^2I + 4\mu J$ on $\ker D$. The matrices $N$ and $N^{\mathsf T}$ permute the orbit indicators, so $\ker D$ is invariant under $U = (H - N - N^{\mathsf T} - cI)/2$ as well; the quotient of $H$ equals $2B_U + 2\Pi + cI$, and the quotient of $J$ equals $\mathbf 1w^{\mathsf T}$.

6. Here $N = P$, $P^3 = I$, $D = P - P^2$ and $N^2 - (N^{\mathsf T})^2 = -D$, so Lemma 5 gives $UD = -DU$. Then $U$ commutes with $D^2 = P + P^2 - 2I = B_3 - 3I$, where $B_3$ is the block-diagonal matrix of ones on the triples; hence for each block of $U$ between triples all row and column sums equal one number $c_{ij} \in \{0, 1, 2, 3\}$: the block equals $0$, $J$, a permutation matrix $Q_0$ or $J - Q_0$. The condition $Q_0D_0 = -D_0Q_0$ for the cycle $P_0$ of a triple excludes the rotations and leaves the transpositions. Inside a triple all pairs are joined by arcs of $\pi$, so the block is zero. The equality for $C_U$ is item 5: the orbits of $\pi^2$ are the triples, $\Pi = I$, $w = 3\mathbf 1$. ∎

**What Theorem 13 closes.** $(14, 5, 4, 1, 2)$, by hand: by items 2 and 4 the cycle lengths are from $\{3, 4, 8\}$, $3a_3 \equiv 2 \pmod 4$ gives $a_3 = 2$, and $[8, 3, 3]$ and $[4, 4, 3, 3]$ remain; a triple needs $3(t - \mu + 1) = 9$ vertices on the other cycles of length divisible by 3, but there are only three. This is the nonexistence proof of [4], obtained anew in [41].

For $(30, 12, 11, 4, 5)$, of the 76 types left by alternation at $\ell = 2$, item 2 and the triple bound of item 3 leave $[3^{10}]$, $[12, 3^6]$, $[12, 12, 3, 3]$ and $[24, 3, 3]$. The type $[12, 3^6]$ is excluded by the Fourier window with equality required at frequency $\pi/6$: there are no partners with sine $-1/2$, and the largest value of the left-hand side over all 32 arrangements is smaller than required. The type $[24, 3, 3]$ is excluded by the edge count of item 4: $24(22 + 3 - 24)/2 = 12 < 3\cdot 2\cdot(11 - 3) = 48$. By item 6, for $[3^{10}]$ the quotient satisfies $C_U^2 + 3C_U - 4I = 15J$, and by the search of the paper [41] its class up to permutation of the triples is unique: it is the Petersen graph on the 2-subsets of $\mathbb Z_5$. Two finite formulas remain (Section 6.7).

The next two sets are the first points of the sign-square lines $q = 7$ and $q = 9$ (Theorem 7, items 2 and 5). For them the cycle types are counted by the program of the supplement. It runs over the partitions of $v$ into cycle lengths left by item 1 and by alternation at $\ell = 2$, that is, into triples and even lengths from 4; to each of them it applies the rest of item 2, the triple bound of item 3, item 4 and the Fourier window; in the window one arrangement $\varepsilon$ of a cycle must satisfy the inequality at all its frequencies and the equality at the frequencies without partners. For $(54, 23, 22, 9, 10)$, of the 1,223 types eight remain: $[24, 24, 3, 3]$, $[12^4, 3, 3]$, $[12^3, 4^3, 3, 3]$, $[12^3, 3^6]$, $[12^2, 4^3, 3^6]$, $[12^2, 3^{10}]$, $[4^3, 3^{14}]$ and $[3^{18}]$, the last being the type of the graph of the paper [45]; the quotient for triples only satisfies $C_U^2 + 3C_U - 10I = 30J$. For $(86, 38, 37, 16, 17)$, with the window checked on cycles of length up to 48, 38 of the 24,808 types remain, and seven of them have cycles of length 60 or 72, on which the window has not been checked.

The window is checked in exact arithmetic. The left-hand side equals $\sum_m C_m\omega^m$ with integer autocorrelations $C_m = \sum_{\ell}\varepsilon_\ell\varepsilon_{\ell + m}$, and the bound is $4(t - \mu) + 3 - (\omega^2 + \omega^{-2})$, so the difference is a polynomial in $\omega$ with integer coefficients. Let $L'$ be the order of the root $\omega$. Equality is decided without rounding: the remainder of the polynomial modulo the cyclotomic polynomial $\Phi_{L'}$, computed with integer coefficients, must be zero. If the remainder is nonzero, then the value at $\omega$ is nonzero too, because $\Phi_{L'}$ is the minimal polynomial of $\omega$; the sign of the value is then found by interval arithmetic, raising the precision until the interval separates from zero. No exclusion rests on rounding.

### 6.7. Computational nonexistence results

For each computation we name the proved reduction, what is searched, and the evidence of the paper in the terms of Section 1.

1. **$(30, 12, 11, 4, 5)$**, papers [39], [41]. The reduction is Theorem 13: relabelling to the canonical $\pi$ of one of the types. In [39] there are 89 SAT instances: 75 types and 14 row normalizations for $[3^{10}]$; the DRUP proofs (Glucose 4 via PySAT), DRAT and LRAT proofs (CaDiCaL 1.9.5) are checked by DRAT-trim and `lrat-check`; an independent CP-SAT model agrees; the controls $(14, 5, 4, 1, 2)$ and $(32, 6, 5, 1, 1)$ are unsatisfiable, and for $(18, 4, 3, 0, 1)$ a graph is found. In [41] there are two formulas, $[3^{10}]$ with a fixed $C_U$ and $[12, 12, 3, 3]$, with 406,816 variables each; DRAT is checked by drat-trim and by the authors' own LRAT checker. The evidence of [39] and [41] is a certificate. The uniqueness of the class of $C_U$ comes from the search of [41], while Theorem 13(6) gives only the equation for $C_U$. The paper [45] does not prove nonexistence: it calls it known, checks the phase condition for the type $[3^{10}]$ on 288 matrices by one CP-SAT program and calls the outcome consistent with the known result.
2. **$(28m, 21m, 18m, 15m, 18m)$ for odd $m$**, paper [19]. The reduction is Theorem 1(c) (a basis of columns that actually occur, and a normal), Theorem 10 (twins at most $4m - 1$, $T \ne n, n + 1$) and the replacement of a basis column at the largest coordinate, which reduces the normal $(1/2)^8$ to two others; for $m = 1$ three normals are possible, for any $m$ five. The search: for $m = 1$ the normal with two zeros is closed by 102 columns, which a dynamic program leaves of 42 candidate orbits, and an integer separating inequality written out in the text, and the normal $(0^4, 1^4)$ by SAT on 87,631 variables with DRAT; for $m = 3$, 72,780 bases and 56 traces of PB proofs (VeriPB) and DRAT. For all odd $m$ the paper gives a separate proof. Forty-five exclusions are closed by Farkas certificates that are checked as identities in $m$; then a separate program checks 3,800,634 exclusions of pairs of normalized rules in integers; the remainder reduces to five SAT formulas in which $m$ does not occur, drat-trim checks their DRAT traces, and one more program checks the formulas and the 24,398 separating inequalities. The traces ($6.7$ GB) are not in the paper's supplement: its program regenerates them, comparing hashes. The evidence of the paper for all odd $m$ is a certificate; the traces were neither regenerated nor checked here. For even $m$ graphs exist: $A \otimes J_\ell$ of the graph on ordered pairs of eight points.
3. **$(36m, 21m, 14m, 11m, 14m)$ for all $m$**, paper [21]. The reduction is Theorem 1(c), Theorem 8 ($\det N_0 = 7h$, $h \le 8$ by the exact $D(8)$), Corollary 3.1 and Theorem 10 ($T \ne n$), and a covering by row bases. The search: 3,032,701 tuples, 302 full-rank slices, 621 pools and 51,030 orthogonal pairs. The evidence is a certificate: for each pool an integer matrix $Y$ with $\langle Y, zb^{\mathsf T}\rangle \ge 0$ and $\langle Y, \cdot\rangle < 0$ on the right-hand side, checked in integers.
4. **Six rays of the paper [15]**: $(72, 21, 7, 4, 7)$, $(100, 35, 14, 9, 14)$, $(81, 63, 56, 47, 56)$, $(96, 80, 75, 65, 75)$, $(81, 24, 8, 5, 8)$, $(50, 35, 28, 23, 28)$ and all their multiples. The reduction is Theorem 1(c): the normal $\alpha = a/\det N_0$ and the pairs $(z_x, \beta_x)$; Theorem 8 bounds $\det N_0$, and Theorem 9 makes the search independent of $m$. Normals with a full-rank slice are excluded by hand or by a search over supports. The numbers of normals are 24, 75, 9, 6, 759 and 63. The evidence is of the kind *two programs*: two or three independent programs agree (C++ and Python, in one case with exact fractions); there are no certificates.
5. **Rank 6 and rank 7 when $t = \mu$**, papers [13] (ranks 6 and 7) and [33] (rank 6). The reduction is Theorems 1(c) and 9 and the integer Farkas criterion on closed collections of pairs. For $g = 5$ the search of [13] gives 729 canonical hyperplanes and 57 ratios $v/k$; the ratios $18/5$ and $24/5$ admit no graph by Theorem 8 without search. For the ratios $12/5$, $9/5$, $8/5$, $7/5$, $14/5$, $18/5$ and $24/5$ nonexistence is formalized in Lean 4 [33]: the search over hyperplanes is checked by `native_decide` in all seven, the cones by the kernel for $12/5$, $14/5$ and $7/5$ and by `native_decide` for $9/5$ and $8/5$, while $18/5$ and $24/5$ have no hyperplanes. The paper [33] closes the ratio $4/3$ by exact Farkas certificates outside Lean and does not recheck rank 7. The Farkas certificates of [13] for the ratio $4/3$ and for the rank-7 ratios $14/3$ and $14/9$ are not given in the text. The search is repeated here for all eight pairs $(g, v/k)$ on which the sets with $v \le 110$ excluded by these papers rest, except the two ratios closed by Theorem 8: $(5, 4/3)$, $(5, 12/5)$, $(5, 9/5)$, $(5, 8/5)$, $(5, 7/5)$, $(5, 14/5)$, $(6, 14/3)$, $(6, 14/9)$. Lemmas 7 and 8 make it finite and small, and Lemma 6 closes without a certificate the hyperplanes on which only unit columns are admissible.

   **Lemma 6.** If, in the notation of Theorem 1(c), every column $\beta_x$ of the matrix $R$ is a unit vector, then $\alpha = (d/\mu)\mathbf 1$.

   *Proof.* The entry $(j, j)$ of the identity (4) equals $\sum_x \beta_x(j)z_x(j)$. For $\beta_x = e_j$ the term equals $z_x(j) = z_x^{\mathsf T}\beta_x = 0$, and for $\beta_x = e_i$, $i \ne j$, it is zero. Hence the diagonal of $RC$ is zero, while by (4) it equals $\mu\alpha_j - d$. ∎

   Hence a hyperplane with a nonconstant normal on which only unit columns are admissible admits no graph. In [13] such hyperplanes were closed by a diagonal certificate; Lemma 6 makes it unnecessary.

   **Lemma 7.** If a zero-one matrix $N$ of order $n$ has a column of ones, then $|\det N| \le D(n - 1)$. Therefore, for $\alpha = M^{-1}\mathbf 1$ with an invertible zero-one matrix $M$ of order $n$ and $\delta = |\det M|$, the vector $a = \delta\alpha$ is an integer vector, $|a_j| \le D(n - 1)$, $\delta \le D(n)$, and $\delta v/k = \mathbf 1^{\mathsf T}a$ is an integer.

   *Proof.* The matrix $K$ of order $n + 1$ with first row and first column of ones and the block $J - 2N$ consists of $\pm 1$, and subtracting the first row from the others gives $\det K = (-2)^n\det N$. If column $j$ of $N$ consists of ones, then column $j + 1$ of $K$ equals $(1, -1, \ldots, -1)^{\mathsf T}$; adding the first column to it gives $(2, 0, \ldots, 0)^{\mathsf T}$, so $|\det K| = 2|\det K'|$ for a $\pm 1$-matrix $K'$ of order $n$. The largest determinant of a $\pm 1$-matrix of order $n$ is $2^{n - 1}D(n - 1)$, whence $|\det N| = |\det K'|/2^{n - 1} \le D(n - 1)$. By Cramer's rule $\alpha_j = \det M^{(j)}/\det M$, where column $j$ of $M^{(j)}$ is replaced by a column of ones; $\delta \le D(n)$ by the definition of $D$. ∎

   **Lemma 8.** Let $H_\alpha = \{z \in \{0, 1\}^n : z^{\mathsf T}\alpha = 1\}$, let $S \subseteq H_\alpha$ be the set of rows of $C$, $B(S) = \{\beta : \mathbf 1^{\mathsf T}\beta = 1,\ z^{\mathsf T}\beta \in \{0, 1\} \text{ for } z \in S\}$ and $\operatorname{cl}(S) = \{z \in H_\alpha : z^{\mathsf T}\beta \in \{0, 1\} \text{ for } \beta \in B(S)\}$. Then $B(\operatorname{cl} S) = B(S)$, and all pairs $(z_x, \beta_x)$ of the graph lie in $P(T) = \{(z, \beta) : z \in T,\ \beta \in B(T),\ z^{\mathsf T}\beta = 0\}$ for $T = \operatorname{cl}(S)$. Therefore, if the system (11) with $w \ge 0$ is unsolvable on $P(T)$ for every normal $\alpha$ and every $T = \operatorname{cl}(S)$ with $S$ of rank $n$, then there is no DSRG $(v, gd, \mu, \mu - d, \mu)$ with these $g$ and $v/k$. The closed set of rows and the inclusion $S \subseteq \operatorname{cl}(S)$ are in Lemma 3.3 of the paper [13]; the equality $B(\operatorname{cl} S) = B(S)$ is proved here, and in this form the lemma directly defines the search.

   *Proof.* From $S \subseteq \operatorname{cl}(S)$ it follows that $B(\operatorname{cl} S) \subseteq B(S)$, and every $\beta \in B(S)$ satisfies $z^{\mathsf T}\beta \in \{0, 1\}$ on $\operatorname{cl}(S)$ by definition. By Theorem 1(c) the rows $z_x$ lie in $S$, the columns $\beta_x$ in $B(S)$, and $z_x^{\mathsf T}\beta_x = 0$. ∎

   Every $\beta \in B(S)$ equals $M^{-1}b$ for a basis $M \subseteq S$ and a binary $b$, so the sets $B(S)$ and $\operatorname{cl}(S)$ are enumerated by a finite search. By Lemma 7 the normals are points of a finite grid: $\delta \le D(n)$, $\delta v/k$ is an integer and $|a_j| \le D(n - 1)$; for each hyperplane the closed collections of rank $n$ are enumerated, and for each of them an integer matrix $Y$ is sought with $\beta^{\mathsf T}Yz \ge 0$ on $P(T)$ and $\langle Y, \tfrac{nk}{v}\alpha\mathbf 1^{\mathsf T} - I\rangle < 0$ (Section 6.2), checked in integers. For the eight pairs $(g, v/k)$, in the same order, there are 39, 7, 20, 19, 17, 1, 31 and 144 hyperplanes, the same numbers as counted by [13]; 24,492, 26, 902, 1,169, 668, 1, 67 and 8,674 closed collections; and 124, 5, 42, 78, 9, 1, 3 and 256 certificates, with entries of absolute value at most 2. For $12/5$, $14/5$, $7/5$ and $14/3$ diagonal certificates $Y = \pm e_je_j^{\mathsf T}$ suffice. For $14/5$ the grid of Lemma 7 leaves 117 candidates $\alpha$, rank 6 is attained by only one hyperplane, and Lemma 6 excludes it. Five of the seven ratios formalized in [33] and the ratio $4/3$ are checked by this search a second time, independently of the formalization and of the paper's audit. The search repeats the nonexistence of 18 pairs with $v \le 110$, including $(40, 30, 27, 21, 27)$, $(80, 60, 54, 42, 54)$ and two pairs with $v = 56$, whose certificates the paper [13] gives only in its supplement, not downloaded here. The certificates, the program that checks them in integers and its negative control are in the supplement.
6. **$(30, 11, 9, 2, 5)$**, paper [43]. The reduction is Theorem 12 ($\sigma = 1$), the anchor triangle (by Lemma 2 of the paper [43], every reciprocal triangle $\Delta$ satisfies $A\mathbf 1_\Delta = \mathbf 1_\Delta + \mathbf 1$ or the same for $A^{\mathsf T}$, that is, by Lemma 3 with $\theta = 1$ each such triangle gives a right or a left eigenvector; the proof of the lemma is an estimate of the two defect vectors $(3I - 3A + J)\mathbf 1_\Delta$ and $\mathbf 1_\Delta^{\mathsf T}(3I - 3A + J)$, which are eigenvectors for $s = -4$, and a count of cells; the lemma is proved in [43], and the proof is checked here; the anchor is such a triangle, made right by transposing the graph if necessary) and Theorem 11(e) with a core of rank 8. The search: an enumeration of cores (710, 979 and 3,136 labelled ones, 249 extensions of rank-6 cores), then 3,769 formulas choosing the types of the outer vertices. The evidence is a certificate: the paper checked every DRAT proof with the program drat-trim, independent enumerators in C++ and Python agree, and the encodings are tested on the graphs $(24, 9, 7, 2, 4)$ and $(36, 13, 11, 2, 6)$. The upper bounds on the multiplicities of the types in the rank-7 interface are not written out in the text, and this step cannot be reproduced from the description in the paper.

## 7. Symmetry exclusions

A symmetry exclusion excludes a class of realizations — graphs with a given group or a given action — but does not settle the pair.

**Theorem 14.** Let $\Gamma$ be a group of automorphisms of a DSRG, and $r, s$ integers.

(a) The character $\psi_r$ of the group $\Gamma$ on $E_r = \ker(A - rI)$ is rational: Galois-conjugate irreducible characters occur in it with equal multiplicities, and the multiplicity of the trivial one equals the multiplicity $f_\Gamma$ of the eigenvalue $r$ of the quotient by the orbits. Therefore, when $|\Gamma|$ is odd, $f \equiv f_\Gamma$ and $g \equiv g_\Gamma \pmod 2$; for a transitive $\Gamma$ of odd order $f$ and $g$ are even.

(b) *Trace lemma.* For an automorphism $\pi_0$ of prime order $p$ with $c_p$ cycles of length $p$, the number $\operatorname{tr}(AP_{\pi_0}) = |\{x : x \to \pi_0(x)\}|$ equals $k + r(f - pa_0) + s(g - pb_0)$ for some integers $a_0, b_0 \ge 0$, $a_0 + b_0 = c_p$, $f - (p - 1)a_0 \ge 0$, $g - (p - 1)b_0 \ge 0$, and $0 \le \operatorname{tr}(AP_{\pi_0}) \le c_pp$.

(c) *Squaring modulo 2.* If an abelian group $P_0$ of odd order acts regularly and $\lambda \equiv \mu \pmod 2$, then there is no Cayley graph on $P_0$ with $0 < t < k$. For a semiregular action with $R_0$ orbits, the sum of the diagonal blocks $D_0 = \sum_iS_{ii} \in \mathbb F_2[P_0]$ satisfies $D_0^2 + cD_0 \equiv R_0(t - \mu)e + R_0\mu\,\mathbf 1_{P_0} \pmod 2$.

(d) *Two orbits.* For a semiregular action of an abelian $G$ of order $v/2$ with two orbits, for every nontrivial character $\chi$ the matrix $\chi(F)$ of order 2 has eigenvalues from $\{r, s\}$, and the multiplicity of $r$ is constant on the Galois orbit of the character; if both eigenvalues equal $\theta$, the block equals $\theta I$.

*Proof.* (a) $E_r$ is defined over $\mathbb Q$ and is preserved by permutation matrices, so $\psi_r(\gamma)$ is a rational algebraic integer, that is, an integer. The multiplicity $\langle\psi_r, \chi\rangle$ is an integer, and a Galois automorphism carries it to $\langle\psi_r, \chi^\varsigma\rangle$. The trivial character occurs with multiplicity equal to the dimension of the $\Gamma$-fixed vectors of $E_r$ — the functions constant on the orbits, on which $A$ acts as the quotient. When $|\Gamma|$ is odd, a nontrivial irreducible $\chi$ is not equal to $\bar\chi$ (Burnside), and complex conjugation is an element of the Galois group, so the nontrivial characters come in pairs. For a transitive $\Gamma$ the quotient is the number $k$, and $f_\Gamma = 0$. (b) The eigenspaces of $A$ are rational and invariant under $P_{\pi_0}$; on a subspace of dimension $D'$ containing $j$ copies of the rational irreducible representation of dimension $p - 1$, the trace of $P_{\pi_0}$ equals $D' - pj$, and there are $c_p$ copies in all. The diagonal of $AP_{\pi_0}$ at $x$ equals $A_{x, \pi_0(x)}$. (c) For $R_0 = 1$, in $\mathbb Z[P_0]$ we have $S_0^2 = (t - \mu)e + (\lambda - \mu)S_0 + \mu P_0$ for the connection set $S_0$. Modulo 2, $S_0^2 \equiv \sum_{x \in S_0}x^2$, and $x \mapsto x^2$ is a bijection of $P_0$. When $\lambda - \mu$ is even, the coefficient of $x \ne e$ on the left equals $[\sqrt x \in S_0]$, and on the right it equals $\mu$. For even $\mu$ this gives $S_0 = \varnothing$, and for odd, $S_0 = P_0 \setminus e$; both contradict $0 < t < k < v - 1$. The general case is the same count for the sum of the diagonal blocks. (d) This is part (a) for $\Gamma = G$ in the form of Theorem 5. ∎

**Theorem 15.** (a) If $\mu \ge 1$ and a prime $p > \max(\kappa, t)$, then every automorphism of order $p$ is semiregular and $p \mid v$. The threshold may be taken for the graph or for its complement.

(b) Let $\gamma$ have order $p$, let $F$ be the nonempty set of its fixed vertices, $\max(\lambda, \mu) < p$, $M_F = A[F, F]$, and let $h_i$ be the number of moved orbits in the reciprocal neighbourhood $U(i)$. Then there are no paths of length 2 between distinct fixed vertices through moved vertices, $M_F^2 + cM_F = \mu J + \operatorname{diag}(t - \mu - ph_i)$ and
$$
\mu(r_i - c_j) = p\,(M_F)_{ij}(h_j - h_i)
$$
for the row sums $r_i$ and the column sums $c_j$ of the matrix $M_F$. If $p \nmid \mu$ and $\lfloor t/p\rfloor < \mu$, then $M_F$ is regular.

(c) If $\Gamma$ is transitive on a set $X$ of $p^a$ points, then its Sylow $p$-subgroup $P$ is transitive on $X$.

*Proof.* (a) Let $\gamma$ of order $p$ fix $x$. The sets $N^+(x)$ of size $\kappa$ and $U(x)$ of size $t$ are invariant, the orbits of $\langle\gamma\rangle$ have size 1 or $p$, and both sets are smaller than $p$; hence all out-neighbours of $x$ are fixed. When $\mu \ge 1$ any two nonadjacent vertices are joined by a path of length 2, the graph is strongly connected, and being fixed propagates to all vertices. (b) A moved intermediate vertex of a path $i \to w \to j$ gives $p$ such paths, whereas there are only $\lambda$ or $\mu$ paths in all. On the diagonal, $t = (M_F^2)_{ii} + ph_i$. $M_F$ commutes with $M_F^2 + cM_F$, and hence with $\mu J + \Delta$, $\Delta = \operatorname{diag}(t - \mu - ph_i)$; the entry $(i, j)$ of the equality $\mu(M_FJ - JM_F) = \Delta M_F - M_F\Delta$ is the formula. When $p \nmid \mu$ it gives $\mu \mid (M_F)_{ij}(h_j - h_i)$, and $0 \le h_i \le \lfloor t/p\rfloor < \mu$ gives $(M_F)_{ij}(h_j - h_i) = 0$. (c) $|\Gamma| = p^a|\Gamma_y|$ for every $y \in X$, so the $p$-part of $|\Gamma_y|$ equals $|P|/p^a$. The stabilizer $P_y = P \cap \Gamma_y$ is a $p$-subgroup of $\Gamma_y$, and $|P_y| \le |P|/p^a$. Hence the orbit of $P$ through $y$ contains $|P : P_y| \ge p^a = |X|$ points, that is, all of $X$. ∎

In part (b) the divisibility $p \nmid \mu$ alone is not enough: the formula gives only $p \mid (r_i - c_j)$; it is the condition $\lfloor t/p\rfloor < \mu$ that makes the right-hand side zero.

**What Theorems 14 and 15 close.** $(27, 7, 4, 1, 2)$ and $(27, 11, 6, 5, 4)$ are not vertex-transitive: $v = 3^3$, by Theorem 15(c) a transitive group contains a transitive Sylow 3-subgroup, whose order is odd, and Theorem 14(a) requires $f, g$ to be even, whereas they are odd. For $(27, 7, 4, 1, 2)$ this is claimed by [28] and [29], for $(27, 11, 6, 5, 4)$ by [29]. The same argument works for every odd $v = p^a$ for which $f$ or $g$ is odd; for $v \le 110$ it excludes vertex transitivity for eight pairs of orders 27, 49 and 81 (Table 6).

$(27, 7, 4, 1, 2)$ has no automorphisms of order $p \ge 5$ (Theorem 15(a)) [28]. Theorem 15(b) reduces an automorphism of order 3 with a nonempty set $F$ of fixed vertices to a single case. With $\mu = 2$ and $\lfloor 4/3\rfloor = 1 < 2$ the matrix $M_F$ is regular of degree $\rho$, and $h_i \in \{0, 1\}$, because a reciprocal neighbourhood of four vertices holds at most one moved triple. The row sums of the equality $M_F^2 + M_F = 2J + \operatorname{diag}(2 - 3h_i)$ give $\rho^2 + \rho = 2|F| + 2 - 3h_i$, so all $h_i$ equal one value $h$. For $h = 1$ the left-hand side is even and the right-hand side odd. For $h = 0$ the matrix $M_F$ is a graph $(|F|, \rho, 4, 1, 2)$; together with $3 \mid |F| \le 24$ (the moved vertices split into triples) this leaves $|F| = 9$, $\rho = 4$, the $3 \times 3$ lattice. Theorem 15 does not exclude this case. The paper [28] closes it by a separate argument: every fixed vertex sends three one-way arcs to one moved triple and receives three from another, and a degree count forces all arcs of $N$ to run along a cycle $F \to R \to S \to F$ of three sets of nine vertices. The paper excludes such a partition of the one-way part in Section 4 by inequalities valid for any graph of the set, without assumptions on the group or on connectivity (Table 6); the evidence of the paper is a proof, not checked here.

$(40, 9, 6, 5, 1)$ has no realization with two orbits of an abelian group of order 20: the paper [22] excludes it by a search over subsets, and here this follows from Theorem 14(d). $(40, 18, 15, 9, 7)$ has no bi-Cayley realizations over abelian groups of order 20 [27], [29]; in [29] this is the case $m = 10$ of the theorem on the family $(4m, 2m - 2, m + 5, m - 1, m - 3)$ (Table 6). $(63, 16, 10, 1, 5)$ and $(81, 22, 12, 3, 7)$ have no semiregular action of an abelian group of odd order with three orbits [29]. The paper [31] excludes automorphisms of order $v/4 \in \{11, 13, 17, 19\}$ for nine sets: the trace lemma gives a window for the number of cycles, and the sieve and the forced completion are carried out by two programs with a positive control on $(28, 13, 7, 6, 6)$. Sieves over two abelian orbits exclude this class for 73 pairs in the paper [22] (listed in the supplement) and for 192 of the 374 pairs examined in the paper [27]. The paper [22] also treats 45 sets in which a semiregular abelian action with $1 + |s|/r$ coclique orbits is admissible. For 39 of them no multiset of row pairs passes the congruences for the quotient; the evidence is one program of the paper. For five more sets the class is excluded explicitly by congruences for the quotient (Table 6): for $(64, 25, 19, 6, 12)$ by a proof, for the other four by a program of the paper; for three of them, $(81, 37, 28, 13, 20)$, $(100, 32, 28, 6, 12)$ and $(100, 41, 29, 12, 20)$, the exclusion is proved here by Theorem 5. The sixth set, $(64, 41, 37, 24, 30)$, realizes this class. The same class is excluded explicitly for two more groups of pairs [27]: for seven pairs with $\mu = 1$ it is forbidden by the bound $\binom{q_{00}}{2} \le \sum_g \lfloor C(g)/2 \rfloor$ on the pairs inside an orbit, and for $(80, 35, 22, 13, 17)$, $(88, 24, 15, 5, 7)$, $(96, 40, 35, 13, 19)$ and $(104, 28, 16, 6, 8)$ by Theorem 14(d) together with a sieve over factor groups of orders from 2 to 5. The evidence for these four pairs is one program of the paper. There are no bi-Cayley realizations over $\mathbb Z_{2p}$ and $D_{2p}$ for nine pairs of orders 28, 44, 76 and 92: the sign block of the quotient by $\mathbb Z_2$ is scalar, and its diagonal $a_0$ comes out half-integral [29]. The same paper excludes the graphs invariant under $\mathbb F_{25} \rtimes \mathbb F_5^\times$ for four more pairs of order 100 by an algebraic argument, congruences on character lines; this argument is not checked here. The paper [25] states that there are no vertex-primitive graphs for any pair with $v \le 110$ that was open at the time of the paper; the evidence is one program without a certificate. The same paper states that seven pairs of orders 33, 35 and 45 have no vertex-transitive graphs: a CP-SAT model enumerates the non-regular minimal transitive groups, the search has no certificate, and for $v = 45$ the minimality of the groups is checked only up to order 200,000. For $(33, 12, 8, 3, 5)$ and $(35, 12, 8, 4, 4)$ the same conclusion follows from Gyürki's list of all vertex-transitive DSRGs on up to 39 vertices, given in the table [2] (Section 2): the list contains no graphs with these sets. In the same way, without a certificate, [25] excludes for two pairs of order 110 the graphs invariant under $\mathrm{AGL}(1, 11)$ on two copies of the 2-subsets of $\mathbb F_{11}$.

By Theorem 15(a), for the pairs of Section 9 automorphisms of prime order not dividing $v$ are impossible above a threshold: $(50, 12, 9, 2, 3)$ from $p \ge 11$, $(54, 14, 12, 2, 4)$ from $p \ge 13$, $(45, 18, 16, 7, 7)$ and $(54, 15, 13, 4, 4)$ from $p \ge 17$; for $(38, 16, 13, 6, 7)$, among the primes $p \ge 17$ only the semiregular order 19 is possible.

## 8. Results

The tables of this section list the published claims of the papers [9]–[46]: constructions of sets that a paper names individually (Table 3), families (Table 4), nonexistence (Table 5) and excluded classes of realizations (Table 6). A row is a pair of a set and its complement, or a family; pairs with $v > 110$ are included if a paper names them. Each row has the two columns of Section 1: the paper's evidence and what is checked here. When a row has several papers, it shows the strongest evidence in the order of Section 1.

### 8.1. Constructions

Table 3 lists 217 sets that the papers name individually, by a quintuple in the title or in the text. The column "Theorem" gives the number of the theorem of Section 4 to which the construction reduces; the entry "1+6" means the switching of Theorem 6 after the reverse product. By the evidence of the papers the rows divide as follows. Graph in the text, 158: of these, 148 graphs are built and checked here, for four sets the proof is checked, and the graphs of orders 126, 294, 346, 458, 586 and 3,927 are not built here. Proof, 4: for all of them it is checked here. Graph in the supplement to the paper, 49, of which 29 are built and checked here. Two programs, 5, of which one graph is built here, $(39, 16, 12, 7, 6)$. One program, 1: this is $(111, 40, 20, 15, 14)$ of the paper [11]. Degenerate sets with $t = k$ (undirected strongly regular graphs) and with $t = 0$ are not included in the table, even when they are seeds of constructions: they are not DSRGs.

Table 4 lists 73 families of constructions: the formula, the range of parameters, the papers, the theorem of Section 4 and both evidence columns. Members of a family that the paper does not name individually are not listed in Table 3. The class of a family is set by the weakest link of its data (Section 1): graph in the text for 43 families, proof for 6, graph in the supplement for 22, one program for 2. For a family, the mark "graph built and checked" means that the graph of at least one of its members is built and checked; the catalogue of the supplement names which. Here the proof is checked for 19 families, the graph of a member is built and checked for 44, and 10 have no check: for nine the data lie only in the supplements of the papers, and for Theorem 7 of the paper [44] the threshold is not computed. Two families, Theorem A of the paper [11] and Theorem 7 of the paper [44], have no members with $v \le 110$.

**Table 3. Constructions: sets that the papers name individually.**

| Set | Complement | Papers | Theorem | Paper's evidence | Checked here |
|---|---|---|---|---|---|
| (18, 4, 3, 0, 1) | (18, 13, 12, 9, 10) | [12] | 2 | graph in the text | graph built and checked |
| (24, 10, 8, 4, 4) | (24, 13, 11, 6, 8) | [30] | 1+6 | proof | proof checked |
| (26, 11, 7, 4, 5) | (26, 14, 10, 7, 8) | [18], [46] | 3, 5 | graph in the text | graph built and checked |
| (30, 13, 8, 5, 6) | (30, 16, 11, 8, 9) | [46] | 3 | graph in the supplement | — |
| (30, 13, 11, 6, 5) | (30, 16, 14, 7, 10) | [23] | 5 | graph in the supplement | graph built and checked |
| (32, 14, 7, 5, 7) | (32, 17, 10, 9, 9) | [9], [16] | 1 | graph in the text | graph built and checked |
| (32, 14, 10, 6, 6) | (32, 17, 13, 8, 10) | [12] | 2 | proof | proof checked |
| (34, 14, 12, 5, 6) | (34, 19, 17, 10, 11) | [46] | 3 | graph in the supplement | graph built and checked |
| (35, 12, 8, 4, 4) | (35, 22, 18, 13, 15) | [27] | 5 | graph in the text | graph built and checked |
| (35, 14, 6, 5, 6) | (35, 20, 12, 11, 12) | [17] | 1 | graph in the text | graph built and checked |
| (36, 16, 14, 8, 6) | (36, 19, 17, 8, 12) | [13], [25] | 5 | graph in the text | graph built and checked |
| (38, 17, 10, 7, 8) | (38, 20, 13, 10, 11) | [46] | 3 | graph in the text | graph built and checked |
| (39, 16, 12, 7, 6) | (39, 22, 18, 11, 14) | [10], [11] | 1 | two programs | graph built and checked |
| (40, 10, 3, 1, 3) | (40, 29, 22, 21, 21) | [17] | 1, 2 | graph in the text | graph built and checked |
| (40, 18, 9, 7, 9) | (40, 21, 12, 11, 11) | [9], [16] | 1, 3 | graph in the text | graph built and checked |
| (40, 18, 12, 8, 8) | (40, 21, 15, 10, 12) | [12], [13], [25] | 1+6, 2, 5 | graph in the text | graph built and checked |
| (40, 18, 15, 9, 7) | (40, 21, 18, 9, 13) | [30] | 1+6 | graph in the text | graph built and checked |
| (42, 12, 9, 4, 3) | (42, 29, 26, 19, 22) | [25] | 5 | graph in the supplement | graph built and checked |
| (42, 15, 9, 4, 6) | (42, 26, 20, 16, 16) | [25] | 5 | graph in the supplement | graph built and checked |
| (42, 19, 11, 8, 9) | (42, 22, 14, 11, 12) | [44], [46] | 1, 3 | graph in the text | graph built and checked |
| (45, 13, 6, 3, 4) | (45, 31, 24, 21, 22) | [35], [36] | 4, 5 | graph in the text | graph built and checked |
| (45, 18, 8, 6, 8) | (45, 26, 16, 15, 15) | [12], [17] | 1, 5 | graph in the text | graph built and checked |
| (46, 21, 12, 9, 10) | (46, 24, 15, 12, 13) | [46] | 3 | graph in the supplement | graph built and checked |
| (48, 19, 14, 5, 9) | (48, 28, 23, 17, 15) | [25] | 5 | graph in the text | graph built and checked |
| (48, 20, 17, 7, 9) | (48, 27, 24, 15, 15) | [40] | 4 | graph in the text | graph built and checked |
| (48, 21, 18, 9, 9) | (48, 26, 23, 13, 15) | [40] | 6 | graph in the text | graph built and checked |
| (48, 22, 11, 9, 11) | (48, 25, 14, 13, 13) | [9], [16] | 1, 3 | graph in the text | graph built and checked |
| (48, 22, 17, 11, 9) | (48, 25, 20, 11, 15) | [25] | 5 | graph in the text | graph built and checked |
| (50, 19, 10, 9, 6) | (50, 30, 21, 16, 21) | [16] | 1 | graph in the text | graph built and checked |
| (51, 18, 10, 5, 7) | (51, 32, 24, 20, 20) | [27] | 5 | graph in the text | graph built and checked |
| (54, 20, 9, 8, 7) | (54, 33, 22, 19, 22) | [16] | 1 | graph in the text | graph built and checked |
| (54, 23, 22, 9, 10) | (54, 30, 29, 16, 17) | [45] | 5 | graph in the text | graph built and checked |
| (54, 24, 17, 10, 11) | (54, 29, 22, 15, 16) | [46] | 3 | graph in the supplement | graph built and checked |
| (56, 13, 7, 6, 2) | (56, 42, 36, 30, 36) | [19] | 4 | graph in the text | graph built and checked |
| (56, 25, 15, 10, 12) | (56, 30, 20, 16, 16) | [12] | 1+6, 2 | graph in the text | graph built and checked |
| (56, 26, 13, 11, 13) | (56, 29, 16, 15, 15) | [9], [16] | 1, 3 | graph in the text | graph built and checked |
| (56, 26, 16, 12, 12) | (56, 29, 19, 14, 16) | [12], [13] | 1+6, 2, 5 | graph in the text | graph built and checked |
| (56, 26, 19, 13, 11) | (56, 29, 22, 13, 17) | [34] | 1+6 | graph in the supplement | graph built and checked |
| (57, 22, 14, 9, 8) | (57, 34, 26, 19, 22) | [10], [11] | 1 | graph in the text | graph built and checked |
| (58, 26, 18, 11, 12) | (58, 31, 23, 16, 17) | [14], [17], [18] | 3, 5 | graph in the text | graph built and checked |
| (58, 27, 15, 12, 13) | (58, 30, 18, 15, 16) | [17] | 5 | graph in the text | graph built and checked |
| (60, 16, 9, 2, 5) | (60, 43, 36, 31, 30) | [42] | 5 | graph in the text | graph built and checked |
| (60, 26, 20, 10, 12) | (60, 33, 27, 18, 18) | [12] | 6 | graph in the supplement | — |
| (62, 28, 19, 12, 13) | (62, 33, 24, 17, 18) | [46] | 3 | graph in the supplement | graph built and checked |
| (62, 29, 16, 13, 14) | (62, 32, 19, 16, 17) | [46] | 3 | graph in the text | graph built and checked |
| (63, 19, 8, 5, 6) | (63, 43, 32, 29, 30) | [35], [36] | 4, 5 | graph in the text | graph built and checked |
| (63, 20, 10, 9, 5) | (63, 42, 32, 26, 32) | [12] | 2 | graph in the text | graph built and checked |
| (63, 23, 10, 9, 8) | (63, 39, 26, 23, 26) | [16] | 1 | graph in the text | graph built and checked |
| (63, 27, 12, 11, 12) | (63, 35, 20, 19, 20) | [17] | 1 | graph in the text | graph built and checked |
| (64, 14, 7, 3, 3) | (64, 49, 42, 37, 39) | [13] | 5 | graph in the text | graph built and checked |
| (64, 17, 8, 3, 5) | (64, 46, 37, 33, 33) | [12], [13], [25] | 2, 5 | graph in the text | graph built and checked |
| (64, 19, 8, 7, 5) | (64, 44, 33, 29, 33) | [16] | 1 | graph in the text | graph built and checked |
| (64, 22, 18, 10, 6) | (64, 41, 37, 24, 30) | [22], [25] | 5 | graph in the text | graph built and checked |
| (64, 28, 14, 10, 14) | (64, 35, 21, 20, 18) | [9], [16] | 1, 4 | graph in the text | graph built and checked |
| (64, 30, 15, 13, 15) | (64, 33, 18, 17, 17) | [9] | 3 | graph in the text | graph built and checked |
| (64, 30, 21, 15, 13) | (64, 33, 24, 15, 19) | [30] | 3 | graph in the supplement | — |
| (65, 25, 10, 9, 10) | (65, 39, 24, 23, 24) | [17] | 1 | graph in the text | graph built and checked |
| (65, 28, 16, 12, 12) | (65, 36, 24, 19, 21) | [27] | 5 | graph in the text | graph built and checked |
| (66, 21, 9, 8, 6) | (66, 44, 32, 28, 32) | [17] | 1 | graph in the text | graph built and checked |
| (66, 30, 20, 13, 14) | (66, 35, 25, 18, 19) | [46] | 3 | graph in the supplement | graph built and checked |
| (66, 31, 17, 14, 15) | (66, 34, 20, 17, 18) | [23], [46] | 2, 3 | graph in the text | graph built and checked |
| (66, 32, 17, 16, 15) | (66, 33, 18, 15, 18) | [17] | 1 | graph in the text | graph built and checked |
| (68, 22, 16, 9, 6) | (68, 45, 39, 28, 33) | [27], [44] | 1, 5 | graph in the text | graph built and checked |
| (69, 23, 8, 7, 8) | (69, 45, 30, 29, 30) | [17] | 1 | graph in the text | graph built and checked |
| (70, 14, 3, 2, 3) | (70, 55, 44, 43, 44) | [17] | 1 | graph in the text | graph built and checked |
| (70, 20, 6, 5, 6) | (70, 49, 35, 34, 35) | [17] | 1 | graph in the text | graph built and checked |
| (70, 27, 12, 11, 10) | (70, 42, 27, 24, 27) | [17] | 1 | graph in the text | graph built and checked |
| (70, 32, 21, 14, 15) | (70, 37, 26, 19, 20) | [46] | 3 | graph in the supplement | graph built and checked |
| (72, 18, 5, 3, 5) | (72, 53, 40, 39, 39) | [17] | 1 | graph in the text | graph built and checked |
| (72, 25, 15, 6, 10) | (72, 46, 36, 30, 28) | [12] | 2 | graph in the text | graph built and checked |
| (72, 26, 11, 10, 9) | (72, 45, 30, 27, 30) | [16] | 1 | graph in the text | graph built and checked |
| (72, 27, 19, 10, 10) | (72, 44, 36, 26, 28) | [12], [14] | 2, 5 | graph in the text | graph built and checked |
| (72, 31, 20, 11, 15) | (72, 40, 29, 23, 21) | [38] | 2, 5 | graph in the text | graph built and checked |
| (72, 32, 23, 13, 15) | (72, 39, 30, 21, 21) | [12], [14] | 5, 6 | graph in the text | graph built and checked |
| (72, 33, 24, 15, 15) | (72, 38, 29, 19, 21) | [12] | 2, 3 | graph in the text | graph built and checked |
| (72, 34, 17, 15, 17) | (72, 37, 20, 19, 19) | [9] | 3 | graph in the text | graph built and checked |
| (72, 34, 23, 17, 15) | (72, 37, 26, 17, 21) | [12] | 3, 6 | graph in the text | graph built and checked |
| (74, 34, 22, 15, 16) | (74, 39, 27, 20, 21) | [46] | 3 | graph in the supplement | graph built and checked |
| (74, 35, 19, 16, 17) | (74, 38, 22, 19, 20) | [17] | 5 | graph in the text | graph built and checked |
| (75, 30, 14, 9, 14) | (75, 44, 28, 27, 24) | [16] | 1 | graph in the supplement | graph built and checked |
| (75, 34, 18, 17, 14) | (75, 40, 24, 19, 24) | [16] | 1 | graph in the supplement | graph built and checked |
| (76, 18, 6, 5, 4) | (76, 57, 45, 42, 45) | [17] | 1 | graph in the text | graph built and checked |
| (76, 19, 5, 4, 5) | (76, 56, 42, 41, 42) | [17] | 1 | graph in the text | graph built and checked |
| (76, 36, 24, 18, 16) | (76, 39, 27, 18, 22) | [30] | 3 | graph in the supplement | graph built and checked |
| (77, 21, 6, 5, 6) | (77, 55, 40, 39, 40) | [17] | 1 | graph in the text | graph built and checked |
| (77, 26, 16, 10, 8) | (77, 50, 40, 31, 35) | [44] | 1 | graph in the text | graph built and checked |
| (77, 34, 16, 15, 15) | (77, 42, 24, 22, 24) | [17] | 1 | graph in the text | graph built and checked |
| (78, 24, 13, 8, 7) | (78, 53, 42, 35, 38) | [27] | 5 | graph in the text | graph built and checked |
| (80, 17, 7, 2, 4) | (80, 62, 52, 48, 48) | [25] | 5 | graph in the supplement | — |
| (80, 18, 8, 4, 4) | (80, 61, 51, 46, 48) | [13] | 5 | graph in the text | graph built and checked |
| (80, 21, 9, 4, 6) | (80, 58, 46, 42, 42) | [12], [13] | 2, 5 | graph in the text | graph built and checked |
| (80, 35, 31, 14, 16) | (80, 44, 40, 24, 24) | [30] | 1+6 | graph in the text | graph built and checked |
| (80, 36, 18, 14, 18) | (80, 43, 25, 24, 22) | [9], [16] | 1, 4 | graph in the text | graph built and checked |
| (80, 36, 32, 16, 16) | (80, 43, 39, 22, 24) | [30] | 1+6 | graph in the text | graph built and checked |
| (80, 38, 19, 17, 19) | (80, 41, 22, 21, 21) | [9] | 3 | graph in the text | graph built and checked |
| (80, 38, 25, 19, 17) | (80, 41, 28, 19, 23) | [30] | 3 | graph in the supplement | graph built and checked |
| (81, 25, 10, 7, 8) | (81, 55, 40, 37, 38) | [36] | 4 | graph in the text | graph built and checked |
| (81, 26, 12, 11, 7) | (81, 54, 40, 34, 40) | [12] | 2 | graph in the text | graph built and checked |
| (81, 29, 12, 11, 10) | (81, 51, 34, 31, 34) | [16] | 1 | graph in the text | graph built and checked |
| (81, 38, 34, 15, 20) | (81, 42, 38, 23, 20) | [25] | 5 | graph in the supplement | — |
| (82, 37, 29, 16, 17) | (82, 44, 36, 23, 24) | [14] | 5 | graph in the text | graph built and checked |
| (82, 38, 24, 17, 18) | (82, 43, 29, 22, 23) | [46] | 3 | graph in the supplement | graph built and checked |
| (82, 39, 21, 18, 19) | (82, 42, 24, 21, 22) | [17] | 5 | graph in the text | graph built and checked |
| (84, 33, 27, 14, 12) | (84, 50, 44, 28, 32) | [44] | 1 | graph in the text | graph built and checked |
| (84, 35, 15, 14, 15) | (84, 48, 28, 27, 28) | [17] | 1 | graph in the text | graph built and checked |
| (84, 37, 23, 14, 18) | (84, 46, 32, 26, 24) | [32], [38] | 2, 3, 6 | graph in the text | graph built and checked |
| (84, 38, 26, 16, 18) | (84, 45, 33, 24, 24) | [40] | 3 | graph in the text | graph built and checked |
| (84, 39, 27, 18, 18) | (84, 44, 32, 22, 24) | [40] | 3, 6 | graph in the text | graph built and checked |
| (84, 40, 26, 20, 18) | (84, 43, 29, 20, 24) | [26], [30] | 3 | graph in the text | graph built and checked |
| (85, 34, 14, 13, 14) | (85, 50, 30, 29, 30) | [17] | 1 | graph in the text | graph built and checked |
| (86, 40, 25, 18, 19) | (86, 45, 30, 23, 24) | [46] | 3 | graph in the supplement | graph built and checked |
| (86, 41, 22, 19, 20) | (86, 44, 25, 22, 23) | [46] | 3 | graph in the text | graph built and checked |
| (87, 29, 10, 9, 10) | (87, 57, 38, 37, 38) | [17] | 1 | graph in the text | graph built and checked |
| (88, 27, 12, 11, 7) | (88, 60, 45, 39, 45) | [16] | 1 | graph in the supplement | — |
| (88, 32, 12, 11, 12) | (88, 55, 35, 34, 35) | [17] | 1 | graph in the text | graph built and checked |
| (88, 41, 23, 18, 20) | (88, 46, 28, 24, 24) | [12] | 1+6, 2 | graph in the text | graph built and checked |
| (88, 42, 21, 19, 21) | (88, 45, 24, 23, 23) | [9] | 3 | graph in the text | graph built and checked |
| (88, 42, 24, 20, 20) | (88, 45, 27, 22, 24) | [12], [13] | 1+6, 2, 5 | graph in the text | graph built and checked |
| (88, 42, 27, 21, 19) | (88, 45, 30, 21, 25) | [26], [30] | 3 | graph in the text | graph built and checked |
| (90, 14, 4, 3, 2) | (90, 75, 65, 62, 65) | [12] | 1, 2 | graph in the text | graph built and checked |
| (90, 27, 17, 8, 8) | (90, 62, 52, 42, 44) | [12] | 2 | graph in the supplement | — |
| (90, 28, 11, 8, 9) | (90, 61, 44, 41, 42) | [12] | 2, 4 | graph in the text | graph built and checked |
| (90, 32, 13, 12, 11) | (90, 57, 38, 35, 38) | [16] | 1 | graph in the text | graph built and checked |
| (90, 33, 21, 12, 12) | (90, 56, 44, 34, 36) | [12] | 2 | graph in the supplement | — |
| (90, 35, 14, 13, 14) | (90, 54, 33, 32, 33) | [17] | 1 | graph in the text | graph built and checked |
| (90, 36, 16, 12, 16) | (90, 53, 33, 32, 30) | [12] | 4 | proof | proof checked |
| (90, 38, 20, 16, 16) | (90, 51, 33, 28, 30) | [44] | 1 | graph in the supplement | — |
| (90, 42, 26, 19, 20) | (90, 47, 31, 24, 25) | [46] | 3 | graph in the supplement | graph built and checked |
| (90, 43, 26, 21, 20) | (90, 46, 29, 22, 25) | [23] | 2 | graph in the supplement | graph built and checked |
| (91, 25, 10, 9, 6) | (91, 65, 50, 45, 50) | [17] | 5 | graph in the text | graph built and checked |
| (91, 26, 8, 6, 8) | (91, 64, 46, 45, 45) | [17] | 1 | graph in the text | proof checked |
| (91, 38, 18, 17, 15) | (91, 52, 32, 28, 32) | [16], [17] | 1, 5 | graph in the text | graph built and checked |
| (91, 39, 18, 15, 18) | (91, 51, 30, 29, 28) | [16], [17] | 1, 5 | graph in the text | graph built and checked |
| (92, 22, 7, 6, 5) | (92, 69, 54, 51, 54) | [17] | 1 | graph in the text | graph built and checked |
| (92, 23, 6, 5, 6) | (92, 68, 51, 50, 51) | [17] | 1 | graph in the text | graph built and checked |
| (92, 44, 28, 22, 20) | (92, 47, 31, 22, 26) | [34] | 3 | graph in the supplement | graph built and checked |
| (93, 34, 18, 13, 12) | (93, 58, 42, 35, 38) | [10], [11] | 1 | graph in the text | graph built and checked |
| (94, 44, 27, 20, 21) | (94, 49, 32, 25, 26) | [46] | 3 | graph in the supplement | graph built and checked |
| (94, 45, 24, 21, 22) | (94, 48, 27, 24, 25) | [46] | 3 | graph in the supplement | graph built and checked |
| (95, 18, 6, 5, 3) | (95, 76, 64, 60, 64) | [17] | 1 | graph in the text | graph built and checked |
| (95, 19, 4, 3, 4) | (95, 75, 60, 59, 60) | [17] | 1 | graph in the text | graph built and checked |
| (95, 37, 16, 15, 14) | (95, 57, 36, 33, 36) | [17] | 1 | graph in the text | proof checked |
| (95, 38, 16, 14, 16) | (95, 56, 34, 33, 33) | [17] | 1 | graph in the text | proof checked |
| (96, 11, 4, 3, 1) | (96, 84, 77, 73, 77) | [12] | 1, 2 | graph in the text | graph built and checked |
| (96, 16, 3, 1, 3) | (96, 79, 66, 65, 65) | [17] | 1 | graph in the text | graph built and checked |
| (96, 17, 4, 3, 3) | (96, 78, 65, 63, 65) | [12] | 2 | graph in the text | graph built and checked |
| (96, 22, 9, 5, 5) | (96, 73, 60, 55, 57) | [12] | 6 | graph in the supplement | — |
| (96, 25, 10, 5, 7) | (96, 70, 55, 51, 51) | [12], [13] | 2, 5 | graph in the text | graph built and checked |
| (96, 26, 11, 7, 7) | (96, 69, 54, 49, 51) | [12] | 2 | graph in the text | graph built and checked |
| (96, 27, 10, 9, 7) | (96, 68, 51, 47, 51) | [16] | 1 | graph in the text | graph built and checked |
| (96, 30, 20, 12, 8) | (96, 65, 55, 42, 48) | [25] | 5 | graph in the supplement | — |
| (96, 36, 15, 11, 15) | (96, 59, 38, 37, 35) | [16] | 1 | graph in the supplement | — |
| (96, 37, 18, 13, 15) | (96, 58, 39, 35, 35) | [12] | 2 | graph in the supplement | — |
| (96, 38, 19, 15, 15) | (96, 57, 38, 33, 35) | [12] | 2 | graph in the text | graph built and checked |
| (96, 39, 18, 17, 15) | (96, 56, 35, 31, 35) | [16] | 1 | graph in the text | graph built and checked |
| (96, 42, 21, 15, 21) | (96, 53, 32, 31, 27) | [9], [16] | 1, 4 | graph in the text | graph built and checked |
| (96, 43, 26, 17, 21) | (96, 52, 35, 29, 27) | [26], [38] | 2, 3 | graph in the text | graph built and checked |
| (96, 44, 22, 18, 22) | (96, 51, 29, 28, 26) | [9], [16] | 1, 4 | graph in the text | graph built and checked |
| (96, 44, 29, 19, 21) | (96, 51, 36, 27, 27) | [12] | 2, 3 | graph in the text | graph built and checked |
| (96, 45, 30, 21, 21) | (96, 50, 35, 25, 27) | [12] | 2, 3 | graph in the text | graph built and checked |
| (96, 46, 23, 21, 23) | (96, 49, 26, 25, 25) | [9] | 3 | graph in the text | graph built and checked |
| (96, 46, 29, 23, 21) | (96, 49, 32, 23, 27) | [12] | 2, 3 | graph in the text | graph built and checked |
| (98, 27, 12, 11, 6) | (98, 70, 55, 48, 55) | [24] | 1 | graph in the text | graph built and checked |
| (98, 34, 16, 15, 10) | (98, 63, 45, 38, 45) | [24] | 1 | graph in the text | graph built and checked |
| (98, 41, 21, 20, 15) | (98, 56, 36, 29, 36) | [16] | 1, 2 | graph in the supplement | — |
| (98, 46, 28, 21, 22) | (98, 51, 33, 26, 27) | [46] | 3 | graph in the supplement | graph built and checked |
| (98, 47, 25, 22, 23) | (98, 50, 28, 25, 26) | [44], [46] | 1, 3 | graph in the text | proof checked |
| (99, 30, 10, 7, 10) | (99, 68, 48, 47, 46) | [16] | 1 | graph in the supplement | — |
| (99, 31, 12, 9, 10) | (99, 67, 48, 45, 46) | [36] | 4 | graph in the text | graph built and checked |
| (99, 35, 14, 13, 12) | (99, 63, 42, 39, 42) | [16] | 1 | graph in the text | graph built and checked |
| (99, 38, 18, 17, 13) | (99, 60, 40, 34, 40) | [24] | 1 | graph in the text | graph built and checked |
| (99, 44, 20, 19, 20) | (99, 54, 30, 29, 30) | [17] | 1 | graph in the text | graph built and checked |
| (100, 30, 18, 11, 8) | (100, 69, 57, 46, 51) | [29] | 1, 5 | graph in the text | graph built and checked |
| (100, 36, 18, 11, 14) | (100, 63, 45, 40, 39) | [12] | 2 | graph in the supplement | — |
| (100, 39, 18, 17, 14) | (100, 60, 39, 34, 39) | [20] | 2 | graph in the supplement | graph built and checked |
| (100, 39, 21, 20, 12) | (100, 60, 42, 32, 42) | [16] | 4 | graph in the text | graph built and checked |
| (100, 44, 22, 21, 18) | (100, 55, 33, 28, 33) | [16] | 1 | graph in the supplement | — |
| (100, 45, 27, 18, 22) | (100, 54, 36, 30, 28) | [29] | 5 | graph in the text | graph built and checked |
| (100, 48, 30, 24, 22) | (100, 51, 33, 24, 28) | [29] | 3, 5 | graph in the text | graph built and checked |
| (102, 17, 3, 2, 3) | (102, 84, 70, 69, 70) | [17] | 1 | graph in the text | graph built and checked |
| (102, 47, 34, 21, 22) | (102, 54, 41, 28, 29) | [46] | 3 | graph in the supplement | graph built and checked |
| (104, 26, 7, 5, 7) | (104, 77, 58, 57, 57) | [17] | 1 | graph in the text | graph built and checked |
| (104, 39, 15, 14, 15) | (104, 64, 40, 39, 40) | [17] | 1 | graph in the text | graph built and checked |
| (104, 49, 27, 22, 24) | (104, 54, 32, 28, 28) | [12] | 1+6, 2 | graph in the text | graph built and checked |
| (104, 50, 25, 23, 25) | (104, 53, 28, 27, 27) | [9] | 3 | graph in the text | graph built and checked |
| (104, 50, 28, 24, 24) | (104, 53, 31, 26, 28) | [12], [13] | 1+6, 2, 5 | graph in the text | graph built and checked |
| (104, 50, 31, 25, 23) | (104, 53, 34, 25, 29) | [30] | 3 | graph in the supplement | graph built and checked |
| (105, 28, 20, 11, 6) | (105, 76, 68, 53, 60) | [44] | 1 | graph in the text | graph built and checked |
| (106, 49, 35, 22, 23) | (106, 56, 42, 29, 30) | [14], [17], [18] | 5 | graph in the text | graph built and checked |
| (106, 50, 30, 23, 24) | (106, 55, 35, 28, 29) | [46] | 3 | graph in the supplement | graph built and checked |
| (106, 51, 27, 24, 25) | (106, 54, 30, 27, 28) | [17] | 5 | graph in the text | graph built and checked |
| (108, 33, 11, 8, 11) | (108, 74, 52, 51, 50) | [9], [16] | 1 | graph in the text | graph built and checked |
| (108, 34, 13, 10, 11) | (108, 73, 52, 49, 50) | [12] | 2, 4 | graph in the text | graph built and checked |
| (108, 37, 19, 10, 14) | (108, 70, 52, 46, 44) | [12] | 2 | graph in the text | graph built and checked |
| (108, 38, 15, 14, 13) | (108, 69, 46, 43, 46) | [16] | 1 | graph in the text | graph built and checked |
| (108, 41, 19, 18, 14) | (108, 66, 44, 38, 44) | [16] | 4 | graph in the text | graph built and checked |
| (108, 45, 20, 17, 20) | (108, 62, 37, 36, 35) | [16] | 1 | graph in the supplement | — |
| (108, 52, 32, 26, 24) | (108, 55, 35, 26, 30) | [12], [13] | 2, 3, 6 | graph in the text | graph built and checked |
| (110, 44, 18, 17, 18) | (110, 65, 39, 38, 39) | [17] | 1 | graph in the text | graph built and checked |
| (110, 52, 31, 24, 25) | (110, 57, 36, 29, 30) | [46] | 3 | graph in the supplement | graph built and checked |
| (111, 40, 20, 15, 14) | (111, 70, 50, 43, 46) | [11] | 1 | one program | — |
| (126, 61, 32, 29, 30) | (126, 64, 35, 32, 33) | [44] | 1, 2 | graph in the text | — |
| (150, 73, 41, 36, 35) | (150, 76, 44, 37, 40) | [23] | 2 | graph in the supplement | graph built and checked |
| (155, 38, 22, 13, 8) | (155, 116, 100, 85, 92) | [11] | 1 | two programs | — |
| (203, 40, 28, 15, 6) | (203, 162, 150, 127, 138) | [11] | 1 | two programs | — |
| (205, 48, 24, 15, 10) | (205, 156, 132, 117, 124) | [11] | 1 | two programs | — |
| (210, 103, 56, 51, 50) | (210, 106, 59, 52, 55) | [23] | 2 | graph in the supplement | — |
| (264, 46, 32, 17, 6) | (264, 217, 203, 176, 189) | [44] | 1 | graph in the text | graph built and checked |
| (294, 145, 74, 71, 72) | (294, 148, 77, 74, 75) | [44] | 1, 2 | graph in the text | — |
| (346, 166, 122, 79, 80) | (346, 179, 135, 92, 93) | [17], [18] | 5 | graph in the text | — |
| (400, 196, 182, 106, 86) | (400, 203, 189, 92, 114) | [30] | 1+6 | graph in the text | graph built and checked |
| (458, 221, 163, 106, 107) | (458, 236, 178, 121, 122) | [17], [18] | 5 | graph in the text | — |
| (497, 82, 34, 21, 12) | (497, 414, 366, 343, 354) | [11] | 1 | two programs | — |
| (586, 284, 210, 137, 138) | (586, 301, 227, 154, 155) | [17], [18] | 5 | graph in the text | — |
| (3927, 202, 104, 54, 8) | (3927, 3724, 3626, 3529, 3577) | [44] | 1 | graph in the text | — |
| (63588, 1005, 756, 257, 12) | (63588, 62582, 62333, 61588, 61835) | [44] | 1 | proof | proof checked |
| (77778, 1190, 308, 160, 16) | (77778, 76587, 75705, 75412, 75558) | [44] | 1 | graph in the supplement | — |

**Table 4. Families of constructions.**

| Family | Parameters | Papers | Theorem | Paper's evidence | Checked here |
|---|---|---|---|---|---|
| $(4rb,\ (2r - 2)b,\ (r - 1)b,\ (r - 3)b,\ (r - 1)b)$ | even $r \ge 8$, $b \ge 1$ | [9] | 1, 3, 4 | graph in the text | graph built and checked |
| $(108 + 27s,\ 33 + 9s,\ 11 + 3s,\ 8 + 3s,\ 11 + 3s)$ | $s \ge 0$ | [9] | 1 | graph in the text | graph built and checked |
| $(mq,\ q + 2m - 3,\ n + 4m - 4,\ n + 2m - 3,\ n + 2)$, $q = 1 + nm$ | $m = 3, 5, 7, 9$ with the set $Z$ found by search; the paper does not prove infiniteness | [10] | 1 | one program | graph built and checked |
| $(mq,\ q + 2m - 3,\ n + 4m - 4,\ n + 2m - 3,\ n + 2)$, $q = 1 + nm$ | even $n$, $4 \le n \le 14$; $q$ a prime power, $q \equiv n + 1 \pmod{2n}$, odd $m \ge 3$; except $(n, q) = (8, 25)$ and $(14, 43)$ | [11] | 1 | one program | graph built and checked |
| $(mq,\ q + 2m - 3,\ n + 4m - 4,\ n + 2m - 3,\ n + 2)$, $q = 1 + nm$ | even $n \ge 4$; $q$ a prime power, $q \equiv n + 1 \pmod{2n}$, $q \ge 3n + 1$ and $q > n^2 2^{r - 1}(r\sqrt q + r + 1)$, $r = (n - 2)/2$; no members with $v \le 110$ | [11] | 1 | proof | proof checked |
| $(mq^2,\ w(q - 1) + q((m - 1)(w - 1) + d),\ w(q - w) + \mu,\ q - 2w + \mu,\ \mu)$, $\mu = m(w - 1)^2 + (w - 1)(2d + 1) + \rho$ | a net of order $q$ with $w + 1$ classes, a seed of order $m$ with numbers $d$, $\rho$ | [12] | 1, 2 | proof | proof checked |
| $(mn^2,\ mn + n - 2,\ 2n + m - 3,\ n + m - 3,\ m + 1)$ and $(mn^2,\ 2mn - 2,\ 2n + 4m - 6,\ n + 4m - 6,\ 4m - 2)$ | $m \ge 2$; $n \ge 2$ in the first form, $n \ge 3$ in the second; the net $\mathbb Z_n^2$ with three classes | [12] | 2 | graph in the text | proof checked |
| $(mh^2p^2,\ hp(hm + 1) - h - 1,\ (h + 1)(hp - h - 1) + mh^2 + h,\ hp + mh^2 - h - 2,\ mh^2 + h)$ | $2 \le h < p$ prime powers, $m \ge 2$ | [12] | 2, 5 | proof | proof checked |
| $(36m,\ 18m - 3,\ 9m + 6,\ 9m - 3,\ 9m - 3)$ | $m \ge 2$ | [12] | 2, 3 | graph in the text | graph built and checked |
| gluing of heterogeneous layers: the sets of the table of Theorem 6 of the paper | at least two layers, seeds with degrees $k_i = \kappa + \beta N_i$ | [12] | 1+6, 2, 3, 5 | graph in the supplement | graph built and checked |
| $(32m,\ 8m + 1,\ 2m + 4,\ 2m - 1,\ 2m + 1)$ | $m \ge 2$ | [12] | 2 | graph in the supplement | — |
| $(36m,\ 12m + 1,\ 4m + 7,\ 4m - 2,\ 4m + 2)$ | $m \ge 2$ | [12] | 2 | graph in the supplement | graph built and checked |
| $(50m,\ 20m - 4,\ 8m + 2,\ 8m - 5,\ 8m - 2)$ | $m \ge 2$ | [12] | 2 | graph in the supplement | — |
| $(48m,\ 12m - 2,\ 3m + 3,\ 3m - 1,\ 3m - 1)$ | $m \ge 2$ | [12] | 6 | graph in the supplement | — |
| $(45m,\ 15m - 3,\ 5m + 7,\ 5m - 2,\ 5m - 2)$ | $m \ge 2$ | [12] | 2 | graph in the supplement | — |
| $(128c - 32,\ 48c - 11,\ 18c,\ 18c - 5,\ 18c - 3)$ | $c \ge 1$ | [12] | 2 | graph in the supplement | — |
| $(45m,\ 18m,\ 8m,\ 6m,\ 8m)$ | $m \ge 1$ | [12] | 1, 4, 5 | graph in the text | proof checked |
| $(18m,\ 6m - 2,\ 2m + 1,\ 2m - 2,\ 2m - 1)$ | $m \ge 2$ | [12] | 2, 4 | graph in the text | graph built and checked |
| $(36m,\ 18m + 1,\ 9m + 8,\ 9m - 1,\ 9m + 3)$ | $m \ge 2$ | [12] | 3, 6 | graph in the supplement | graph built and checked |
| $(36m,\ 18m + 3,\ 9m + 12,\ 9m + 3,\ 9m + 3)$ | $m \ge 2$ | [12] | 3, 5, 6 | graph in the text | graph built and checked |
| $(8m,\ 4m - 2,\ 2m + 2,\ 2m - 2,\ 2m - 2)$ | odd $m \ge 3$ | [13] | 1+6, 2, 5 | graph in the text | graph built and checked |
| $(16m,\ 4m + 1,\ m + 4,\ m - 1,\ m + 1)$ | $m \ge 2$ | [13] | 2, 5 | graph in the text | graph built and checked |
| $(36(2u + 1),\ 36u + 16,\ 18u + 14,\ 18u + 8,\ 18u + 6)$ | $u \ge 0$ | [13] | 2, 3, 5 | graph in the text | graph built and checked |
| $(50c,\ 30c,\ 21c,\ 16c,\ 21c)$ | $c \ge 1$ | [16] | 1, 4 | graph in the text | graph built and checked |
| $(75c,\ 30c,\ 14c,\ 9c,\ 14c)$ | $c \ge 1$ | [16] | 1 | graph in the supplement | graph built and checked |
| $(98c,\ 56c,\ 36c,\ 29c,\ 36c)$ | $c \ge 1$ | [16] | 1, 2 | graph in the supplement | — |
| $(96c,\ 36c,\ 15c,\ 11c,\ 15c)$ | $c \ge 1$ | [16] | 1 | graph in the supplement | — |
| $(9rc,\ 3(2r - 1)c,\ 2(2r - 1)c,\ (4r - 5)c,\ 2(2r - 1)c)$ | $6 \le r \le 12$, $c \ge 1$ | [16] | 1, 4 | graph in the text | graph built and checked |
| $(75c,\ 40c,\ 24c,\ 19c,\ 24c)$ | $c \ge 1$ | [16] | 1 | graph in the supplement | graph built and checked |
| $(64c,\ 44c,\ 33c,\ 29c,\ 33c)$ | $c \ge 1$ | [16] | 1 | graph in the text | graph built and checked |
| $(100c,\ 55c,\ 33c,\ 28c,\ 33c)$ | $c \ge 1$ | [16] | 1 | graph in the supplement | — |
| $(96c,\ 68c,\ 51c,\ 47c,\ 51c)$ | $c \ge 1$ | [16] | 1 | graph in the text | graph built and checked |
| $(rqk/p,\ k,\ \mu,\ \mu - a,\ \mu)$, $k = a(rH - 1)$, $\mu = paH/q$, $H = d(q - 1) + 1$ | $1 \le p < q$, $d$, $a$, $r$ with the divisibility conditions of the paper | [17] | 1, 2, 5 | graph in the text | proof checked |
| $(v,\ k,\ \mu,\ \mu - 1,\ \mu)$ | $k(k + 1) = \mu v$, $0 < \mu < k < v$ | [17] | 1, 5 | graph in the text | graph built and checked |
| $((a + 1)n,\ an,\ a^2h,\ a^2h - a,\ a^2h)$, $n = (a + 1)h - 1$ | $a \ge 2$, $h \ge a$ | [17] | 1 | graph in the text | proof checked |
| $(q(qh - 1),\ 2(qh - 1),\ 4h,\ 4h - 2,\ 4h)$ | $q \ge 3$, $h \ge 2$ | [17] | 1, 5 | graph in the text | proof checked |
| $(2P,\ P - 1 - a,\ (P - 1)/2 + a^2,\ (P - 3)/2 - a,\ (P - 1)/2 - a)$ | $P = (2a + 1)^2 + 4$ prime | [17] | 1, 3, 5 | graph in the text | graph built and checked |
| $(2n,\ n - 2,\ (n + 1)/2,\ (n - 5)/2,\ (n - 3)/2)$ | $n = r^2 + (r + d)^2$: $d = 1$, $r \ge 2$ or $d = 3$, $r \ge 1$, $3 \nmid r$ | [17] | 3, 5 | graph in the text | proof checked |
| $(2n,\ n - 2,\ (n + 1)/2,\ (n - 5)/2,\ (n - 3)/2)$ | $n = b^2 + 1$ for even $b$, $n = b^2 + 4$ for odd, $b \ge 3$ | [17] | 3, 5 | graph in the text | proof checked |
| $(q(2q - 1),\ a(2q - 1),\ 2a^2,\ 2a^2 - a,\ 2a^2)$ | $1 \le a$, $3a \le 2q$ | [17] | 1, 5 | graph in the text | proof checked |
| $(70 + 25u,\ 42 + 15u,\ 27 + 9u,\ 24 + 9u,\ 27 + 9u)$ | $u \ge 0$ | [17] | 1 | graph in the text | proof checked |
| $(77 + 121u,\ 42 + 66u,\ 24 + 36u,\ 22 + 36u,\ 24 + 36u)$ | $u \ge 0$ | [17] | 1 | graph in the text | graph built and checked |
| $(2P,\ P - 1 - a,\ (P - 1)/2 + a^2,\ (P - 3)/2 - a,\ (P - 1)/2 - a)$ | $P = (2a + 1)^2 + 4$ prime, $a \ge 1$; the infiniteness of the family is equivalent to the infiniteness of such primes | [18] | 3, 5 | graph in the text | graph built and checked |
| $(28m,\ 21m,\ 18m,\ 15m,\ 18m)$ | even $m \ge 2$; for odd $m$ there is no graph (Table 5) | [19] | 4 | graph in the text | graph built and checked |
| $(2mu^2,\ mu(u + 1),\ 2ma^2 + a,\ 2ma^2 + a - u,\ 2ma^2 + a)$, $a = (u + 1)/2$ | odd $u \ge 3$, $m \ge 1$ | [20] | 1, 2 | graph in the supplement | graph built and checked |
| $(30(2u + 1),\ 30u + 13,\ 15u + 11,\ 15u + 6,\ 15u + 5)$ | $u \ge 1$ | [23] | 2 | graph in the supplement | graph built and checked |
| $(98m^2,\ 35m - 1,\ 7m + 9,\ 7m + 8,\ 10)$ | $m \ge 1$ | [24] | 1 | graph in the text | graph built and checked |
| $(2u^2,\ 4u - 1,\ u + 5,\ u + 4,\ 6)$ | $u = 4$ and $u \ge 6$ | [24] | 1 | graph in the text | graph built and checked |
| $a(99 + 27m,\ 60 + 18m,\ 40 + 12m,\ 34 + 12m,\ 40 + 12m)$ | $a \ge 1$, $m \ge 0$ | [24] | 1 | graph in the text | graph built and checked |
| $(98a,\ 63a,\ 45a,\ 38a,\ 45a)$ | $a \ge 1$ | [24] | 1 | graph in the text | graph built and checked |
| $(98a,\ 70a,\ 55a,\ 48a,\ 55a)$ | $a \ge 1$ | [24] | 1 | graph in the text | graph built and checked |
| $(N,\ N/2 - 2,\ N/4 + 5,\ N/4 - 1,\ N/4 - 3)$ | $N = 36a + 48b$, $a, b \ge 0$, $a + b \ge 1$; $N = 36$ and $48$ are the seeds | [26] | 3 | graph in the text | proof checked |
| $(28 + 60m,\ 12 + 30m,\ 12 + 15m,\ 6 + 15m,\ 4 + 15m)$ | $m \ge 1$ | [26] | 3 | graph in the text | graph built and checked |
| $(N,\ N/2 - 5,\ N/4 + 2,\ N/4 - 7,\ N/4 - 3)$ | $N = 48m$, $m \ge 1$ | [26] | 2, 3 | graph in the text | graph built and checked |
| $(N,\ N/2 - 2,\ N/4 + 5,\ N/4 - 1,\ N/4 - 3)$ | $N = 36\alpha + 40\beta + 48\gamma + 60\delta + 64\varepsilon$ with a positive sum of coefficients; this includes all $N \equiv 0 \pmod 4$, $N \ge 96$ | [30] | 2, 3, 5, 6 | graph in the supplement | graph built and checked |
| $(hm,\ h(m - 1)/2,\ h(m + h - 2)/4,\ h(m - 2)/4,\ h(m - 2)/4)$ and $(hm,\ h(m + 1)/2,\ h(m + h + 2)/4,\ h(m + 2)/4,\ h(m + 2)/4)$ | $4 \mid h$, a Hadamard matrix of order $h$, even $m \ge h$ | [30] | 1+6, 2, 5 | proof | proof checked |
| $(N,\ N/2 - 5,\ N/4 + 2,\ N/4 - 7,\ N/4 - 3)$ | $N = 36 + 48h$, $h \ge 1$ | [32] | 3 | graph in the supplement | graph built and checked |
| $(N,\ N/2 - 2,\ N/4 + 5,\ N/4 - 1,\ N/4 - 3)$ | $N \equiv 0 \pmod 4$, $N \ge 72$ | [34] | 3 | graph in the supplement | proof checked |
| $(9q,\ 3q - 2,\ q + 1,\ q - 2,\ q - 1)$ | $q \ge 5$ | [36] | 2, 4 | graph in the text | graph built and checked |
| $(12m,\ 6m - 5,\ 3m + 2,\ 3m - 7,\ 3m - 3)$ | $m \ge 3$: for $m = 3, 4, 5$ known seeds, for $m \ge 6$ a gluing of seeds of orders 36, 48 and 72 | [38] | 2, 3, 5, 6 | proof | graph built and checked |
| $(N,\ N/2 - 4,\ N/4 + 5,\ N/4 - 5,\ N/4 - 3)$ | $N = 36a + 48b > 36$ | [40] | 2, 3, 4 | graph in the text | graph built and checked |
| $(N,\ N/2 - 3,\ N/4 + 6,\ N/4 - 3,\ N/4 - 3)$ | $N = 36a + 48b > 36$ | [40] | 2, 3, 6 | graph in the text | graph built and checked |
| $(60(3u + 1),\ 60u + 16,\ 20u + 9,\ 20u + 2,\ 20u + 5)$ | $u \ge 0$ | [42] | 5 | graph in the text | graph built and checked |
| $(h(2h(q - 1) + 1),\ 2hq - 2,\ 4h - 6 + 2q,\ 2h - 5 + 2q,\ 2q)$ | $h \ge 2$, $q \ge 3$ | [44] | 1, 5 | graph in the text | proof checked |
| $(a(9a + 1),\ 12a - 3,\ 9a,\ 3a + 5,\ 12)$, $a = 3(27^D - 1)/26$ | $D \ge 1$ | [44] | 1 | graph in the text | proof checked |
| $(a(as^2 + 1),\ as(s + 1) - s,\ as^2,\ as + s^2 - s - 1,\ s(s + 1))$ | fixed $s \ge 2$ and $a$ above a threshold that the paper does not give | [44] | 1 | proof | — |
| $(2M,\ M - 2,\ (M + 1)/2,\ (M - 5)/2,\ (M - 3)/2)$ | $M = 21(2u + 1)$ or $49(2u + 1)$, $u \ge 0$ | [44] | 1, 2 | graph in the text | graph built and checked |
| $(m(m - 1)/6,\ 4(m - 1)/3 - 2,\ 2(m - 1)/3 + 2,\ (m - 1)/3 + 3,\ 8)$ | $m = 22 \cdot 7^j$, $j \ge 0$ | [44] | 1 | graph in the text | graph built and checked |
| $(m(m - 1)/14,\ 8(m - 1)/7 - 2,\ 2(m - 1)/7 + 10,\ (m - 1)/7 + 11,\ 16)$ | $m = 36 \cdot 29^j$, $j \ge 0$ | [44] | 1 | graph in the supplement | — |
| $(N,\ (N - 4)/2,\ (N + 2)/4,\ (N - 10)/4,\ (N - 6)/4)$ | $N \equiv 2 \pmod{12}$, $N \ge 26$ | [46] | 3, 5 | graph in the text | proof checked |
| $(N,\ (N - 4)/2,\ (N + 2)/4,\ (N - 10)/4,\ (N - 6)/4)$ | $N \equiv 6, 10 \pmod{12}$, $N \ge 30$; the seeds of orders 30 and 34 from the supplement of the paper | [46] | 3 | graph in the supplement | proof checked |
| $(N,\ (N - 6)/2,\ (N + 14)/4,\ (N - 14)/4,\ (N - 10)/4)$ | $N \equiv 2 \pmod 4$, $N \ge 54$ | [46] | 3 | graph in the supplement | graph built and checked |
| $(102, 47, 34, 21, 22)$ and the points $N = 50 + 52j$ of the line $q = 7$ | $j \ge 1$ | [46] | 3 | graph in the supplement | graph built and checked |

### 8.2. Nonexistence

Table 5 lists 36 pairs with nonexistence and rows with a formula, 19 families of nonexistence: rays of multiple sets and the line of the paper [43]. Four pairs are proved here: $(14, 5, 4, 1, 2)$ by Theorem 13, and $(54, 15, 5, 2, 5)$, $(96, 20, 5, 1, 5)$ and $(108, 30, 10, 4, 10)$ by Theorem 8. For eighteen more pairs the paper's certificate is repeated here by a search with integer Farkas certificates (Section 6.7, item 5). For the remaining fourteen the paper's evidence is a certificate (nine pairs) or two programs (five pairs), and it is not repeated here. Of the eighteen rays, two are closed here by Theorem 8 and eight are repeated by the search of Section 6.7; for the other eight the paper's evidence is a certificate (four rays) or two programs (four rays). The line of the paper [43] is closed by the rank for $d = 1$ and by Duval's inequality for $d \ge 7$.

**Table 5. Nonexistence: pairs, papers, argument and evidence.**

| Set | Complement | Papers | Argument and computation | Paper's evidence | Checked here |
|---|---|---|---|---|---|
| (14, 5, 4, 1, 2) | (14, 8, 7, 4, 5) | [41] | Theorem 13 (cycle types), by hand | proof | proved here |
| (24, 10, 5, 3, 5) | (24, 13, 8, 7, 7) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (27, 11, 6, 5, 4) | (27, 15, 10, 7, 10) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (28, 6, 3, 2, 1) | (28, 21, 18, 15, 18) | [19] | Theorems 1(c), 10; SAT, DRAT | certificate | — |
| (30, 11, 9, 2, 5) | (30, 18, 16, 11, 10) | [43] | Theorems 11, 12; core and 3,769 formulas, DRAT | certificate | — |
| (30, 12, 11, 4, 5) | (30, 17, 16, 9, 10) | [39], [41] | Theorem 13; SAT, DRAT and LRAT | certificate | — |
| (32, 11, 6, 5, 3) | (32, 20, 15, 11, 15) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (36, 14, 7, 6, 5) | (36, 21, 14, 11, 14) | [21] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| (40, 9, 6, 5, 1) | (40, 30, 27, 21, 27) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates | certificate | search repeated |
| (48, 20, 10, 6, 10) | (48, 27, 17, 16, 14) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (49, 13, 8, 7, 2) | (49, 35, 30, 23, 30) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (50, 14, 7, 6, 3) | (50, 35, 28, 23, 28) | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| (54, 15, 5, 2, 5) | (54, 38, 28, 27, 26) | [13], [33] | Theorem 8 (denominator); Lean [33] | certificate | proved here |
| (54, 23, 13, 12, 8) | (54, 30, 20, 14, 20) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (56, 12, 3, 1, 3) | (56, 43, 34, 33, 33) | [13] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates | certificate | search repeated |
| (56, 19, 10, 9, 5) | (56, 36, 27, 21, 27) | [13] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates | certificate | search repeated |
| (64, 23, 13, 12, 6) | (64, 40, 30, 22, 30) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (72, 21, 7, 4, 7) | (72, 50, 36, 35, 34) | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| (72, 29, 15, 14, 10) | (72, 42, 28, 22, 28) | [21] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| (72, 30, 15, 9, 15) | (72, 41, 26, 25, 21) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (80, 19, 13, 12, 2) | (80, 60, 54, 42, 54) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates | certificate | search repeated |
| (81, 17, 10, 9, 2) | (81, 63, 56, 47, 56) | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| (81, 24, 8, 5, 8) | (81, 56, 40, 39, 38) | [15] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| (81, 35, 20, 19, 12) | (81, 45, 30, 21, 30) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (84, 20, 11, 10, 3) | (84, 63, 54, 45, 54) | [19] | Theorems 1(c), 10; SAT and PB, DRAT | certificate | — |
| (96, 15, 10, 9, 1) | (96, 80, 75, 65, 75) | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| (96, 20, 5, 1, 5) | (96, 75, 60, 59, 57) | [13], [33] | Theorem 8 (denominator); Lean [33] | certificate | proved here |
| (96, 35, 20, 19, 9) | (96, 60, 45, 33, 45) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (96, 40, 20, 12, 20) | (96, 55, 35, 34, 28) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (98, 27, 17, 16, 4) | (98, 70, 60, 46, 60) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| (98, 35, 15, 8, 15) | (98, 62, 42, 41, 36) | [13], [33] | Theorems 1(c), 9, Lemmas 6, 7; search over 117 candidates $\alpha$; Lean [33] | certificate | search repeated |
| (100, 29, 15, 14, 6) | (100, 70, 56, 46, 56) | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| (100, 35, 14, 9, 14) | (100, 64, 43, 42, 39) | [15] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| (108, 30, 10, 4, 10) | (108, 77, 57, 56, 52) | [13], [33] | Theorem 8 (denominator); Lean [33] | certificate | proved here |
| (108, 44, 23, 22, 15) | (108, 63, 42, 33, 42) | [21] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| (108, 47, 27, 26, 16) | (108, 60, 40, 28, 40) | [13], [33] | Theorems 1(c), 9, Lemmas 7, 8; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| $(24m,\ 10m,\ 5m,\ 3m,\ 5m)$, $m \ge 1$ | $(24m,\ 14m - 1,\ 9m - 1,\ 9m - 2,\ 7m)$ | [13], [33] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| $(27m,\ 15m,\ 10m,\ 7m,\ 10m)$, $m \ge 1$ | $(27m,\ 12m - 1,\ 7m - 1,\ 7m - 2,\ 4m)$ | [13], [33] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| $(32m,\ 20m,\ 15m,\ 11m,\ 15m)$, $m \ge 1$ | $(32m,\ 12m - 1,\ 7m - 1,\ 7m - 2,\ 3m)$ | [13], [33] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| $(40m,\ 30m,\ 27m,\ 21m,\ 27m)$, $m \ge 1$ | $(40m,\ 10m - 1,\ 7m - 1,\ 7m - 2,\ m)$ | [13], [33] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates | certificate | search repeated |
| $(49m,\ 35m,\ 30m,\ 23m,\ 30m)$, $m \ge 1$ | $(49m,\ 14m - 1,\ 9m - 1,\ 9m - 2,\ 2m)$ | [13], [33] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| $(54m,\ 15m,\ 5m,\ 2m,\ 5m)$, $m \ge 1$ | $(54m,\ 39m - 1,\ 29m - 1,\ 29m - 2,\ 26m)$ | [13], [33] | Theorem 8 (denominator); Lean [33] | certificate | proved here |
| $(96m,\ 20m,\ 5m,\ m,\ 5m)$, $m \ge 1$ | $(96m,\ 76m - 1,\ 61m - 1,\ 61m - 2,\ 57m)$ | [13], [33] | Theorem 8 (denominator); Lean [33] | certificate | proved here |
| $(98m,\ 35m,\ 15m,\ 8m,\ 15m)$, $m \ge 1$ | $(98m,\ 63m - 1,\ 43m - 1,\ 43m - 2,\ 36m)$ | [13], [33] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates; Lean [33] | certificate | search repeated |
| $(56m,\ 12m,\ 3m,\ m,\ 3m)$, $m \ge 1$ | $(56m,\ 44m - 1,\ 35m - 1,\ 35m - 2,\ 33m)$ | [13] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates | certificate | search repeated |
| $(56m,\ 36m,\ 27m,\ 21m,\ 27m)$, $m \ge 1$ | $(56m,\ 20m - 1,\ 11m - 1,\ 11m - 2,\ 5m)$ | [13] | Theorems 1(c), 9; search over hyperplanes, Farkas certificates | certificate | search repeated |
| $(72m,\ 21m,\ 7m,\ 4m,\ 7m)$, $m \ge 1$ | $(72m,\ 51m - 1,\ 37m - 1,\ 37m - 2,\ 34m)$ | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| $(100m,\ 35m,\ 14m,\ 9m,\ 14m)$, $m \ge 1$ | $(100m,\ 65m - 1,\ 44m - 1,\ 44m - 2,\ 39m)$ | [15] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| $(81m,\ 63m,\ 56m,\ 47m,\ 56m)$, $m \ge 1$ | $(81m,\ 18m - 1,\ 11m - 1,\ 11m - 2,\ 2m)$ | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| $(96m,\ 80m,\ 75m,\ 65m,\ 75m)$, $m \ge 1$ | $(96m,\ 16m - 1,\ 11m - 1,\ 11m - 2,\ m)$ | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| $(81m,\ 24m,\ 8m,\ 5m,\ 8m)$, $m \ge 1$ | $(81m,\ 57m - 1,\ 41m - 1,\ 41m - 2,\ 38m)$ | [15] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| $(50m,\ 35m,\ 28m,\ 23m,\ 28m)$, $m \ge 1$ | $(50m,\ 15m - 1,\ 8m - 1,\ 8m - 2,\ 3m)$ | [15] | Theorems 1(c), 8, 9; search over normals, two programs | two programs | — |
| $(28m,\ 21m,\ 18m,\ 15m,\ 18m)$, odd $m$ | $(28m,\ 7m - 1,\ 4m - 1,\ 4m - 2,\ m)$ | [19] | Theorems 1(c), 10; SAT, DRAT | certificate | — |
| $(36m,\ 21m,\ 14m,\ 11m,\ 14m)$, $m \ge 1$ | $(36m,\ 15m - 1,\ 8m - 1,\ 8m - 2,\ 5m)$ | [21] | Theorems 1(c), 8, 9, 10; integer separators | certificate | — |
| $(6d,\ 2d + 1,\ 2d - 1,\ 2,\ d)$, $d = 1$ and $d \ge 7$ | $(6d,\ 4d - 2,\ 4d - 4,\ 3d - 4,\ 2d)$ | [43] | rank for $d = 1$; Duval's inequality $\mu - \lambda \le 2(k - t)$ for $d \ge 7$ | proof | — |

### 8.3. Excluded classes of realizations

Table 6 lists the exclusions of classes of realizations in 49 rows; two of them are exclusions for whole families of sets, of the papers [9] and [29]. An exclusion does not settle a pair but narrows the search: a search with a prescribed automorphism group within an excluded class is useless. The large-scale sieves of the papers [22], [25] and [27] are recorded in one row each. The explicit exclusions of the papers, and the exclusions given by the theorems of this survey ("this survey" in the column "Papers"), are recorded in one row per pair or per group of pairs with a common argument.

**Table 6. Exclusions of classes of realizations.**

| Set or pairs | Papers | Excluded class | Argument | Paper's evidence | Checked here |
|---|---|---|---|---|---|
| (27, 7, 4, 1, 2) | [28], [29] | vertex-transitive | Theorems 14(a), 15(c) | proof | proof checked |
| (27, 7, 4, 1, 2) | [28] | automorphisms of prime order p ≥ 5 | Theorem 15(a) | proof | proof checked |
| (27, 7, 4, 1, 2) | [28] | automorphism of order 3 with fixed vertices | Theorem 15(b); the case of nine fixed vertices — the paper's argument via the exclusion of three parts with a cyclic course of arcs | proof | — |
| (27, 7, 4, 1, 2) | [28] | one-way part is a Cayley graph (five groups of order 27) | Theorem 11; 207 formulas, DRAT | certificate | — |
| (27, 11, 6, 5, 4) | [29] | vertex-transitive | Theorems 14(a), 15(c) | proof | proof checked |
| (40, 18, 15, 9, 7) | [27], [29] | bi-Cayley over abelian groups of order 20 | Theorem 14(d) | proof | proof checked |
| (40, 18, 15, 9, 7) | [29] | bi-Cayley over Dic₅, D₂₀, F₂₀ | search, program of the paper | one program | — |
| (40, 9, 6, 5, 1) | [22] | two orbits over an abelian group of order 20 | Theorem 14(d) in place of the paper's search | one program | proved here |
| (44, 12, 8, 1, 4) | [31] | automorphism of order 11 | Theorem 14(b); sieve, two programs | two programs | — |
| (52, 18, 15, 8, 5) | [31] | automorphism of order 13 | Theorem 14(b); sieve, two programs | two programs | — |
| (76, 17, 13, 6, 3) | [31] | automorphism of order 19 | Theorem 14(b); sieve, two programs | two programs | — |
| (44, 20, 16, 10, 8) | [31] | automorphism of order 11 | Theorem 14(b); sieve, two programs | two programs | — |
| (52, 24, 18, 12, 10) | [31] | automorphism of order 13 | Theorem 14(b); sieve, two programs | two programs | — |
| (68, 32, 22, 16, 14) | [31] | automorphism of order 17 | Theorem 14(b); sieve, two programs | two programs | — |
| (44, 17, 13, 4, 8) | [31] | automorphism of order 11 | Theorem 14(b); sieve, two programs | two programs | — |
| (52, 21, 15, 6, 10) | [31] | automorphism of order 13 | Theorem 14(b); sieve, two programs | two programs | — |
| (68, 29, 19, 10, 14) | [31] | automorphism of order 17 | Theorem 14(b); sieve, two programs | two programs | — |
| (63, 16, 10, 1, 5) | [29] | semiregular action of an abelian group of odd order with three orbits | Theorem 14(c) | proof | proof checked |
| (81, 22, 12, 3, 7) | [29] | semiregular action of an abelian group of odd order with three orbits | Theorem 14(c) | proof | proof checked |
| (35, 12, 8, 4, 4) | [27] | Cayley graph of an abelian group of odd order (the set is constructed otherwise, Table 3) | Theorem 14(c) | not described | proved here |
| 73 pairs (Corollary 4.2 of [22]; listed in the supplement to this paper) | [22] | two orbits over an abelian group of order v/2 | Theorem 14(d) in place of the paper's sieve | one program | proved here |
| 192 of 374 pairs | [27] | two orbits over abelian groups | Theorem 14(d); census, program of the paper | one program | — |
| all pairs of the paper's list with $v \le 110$ | [25] | vertex-primitive | search over orbitals, one program | one program | — |
| 39 pairs | [22] | semiregular abelian action with $1 + \lvert s\rvert/r$ coclique orbits | Theorem 5: congruences for the quotient; search over multisets of row pairs | one program | — |
| (27, 7, 4, 1, 2) | [28] | one-way part split into three parts with a cyclic course of arcs | inequalities of Section 4 of the paper | proof | — |
| (27, 7, 4, 1, 2) | [28] | automorphism of order 3 without fixed vertices | search over quotients by nine orbits with an independent check of the survivors | one program | — |
| (27, 7, 4, 1, 2) | [28] | involutions with 1, 3, 5, 7, 9, 23 or 25 fixed vertices | idempotents over $\mathrm{GF}(2)$, for $f = 7, 9$ a search | one program | — |
| (27, 7, 4, 1, 2), (27, 11, 6, 5, 4) | [29] | semiregular action with three orbits of 9 vertices | search, program of the paper | one program | — |
| (27, 7, 4, 1, 2), (27, 11, 6, 5, 4) | [29] | action with orbits 9 and 18 and a Sylow 3-subgroup of order 9 | quotient $B$ by the orbits 9 and 18 and the orbits of the Sylow subgroup on the 18-orbit | proof | — |
| (28, 6, 3, 2, 1), (66, 10, 5, 4, 1), (70, 9, 3, 2, 1), (88, 9, 3, 0, 1), (96, 10, 5, 1, 1), (96, 11, 4, 3, 1), (104, 11, 7, 2, 1) | [27] | two orbits over an abelian group of order v/2 | pair bound $\binom{q_{00}}{2} \le \sum_g \lfloor C(g)/2 \rfloor$ for $\mu = 1$ | proof | proof checked |
| (28, 6, 3, 2, 1), (44, 12, 8, 1, 4), (76, 17, 13, 6, 3), (76, 18, 6, 5, 4), (76, 20, 10, 3, 6), (92, 21, 14, 7, 4), (92, 22, 7, 6, 5), (92, 24, 11, 4, 7), (92, 25, 22, 3, 8) | [29] | bi-Cayley over $\mathbb Z_{2p}$ and $D_{2p}$ | the sign block of the $\mathbb Z_2$ quotient is scalar, $a_0$ is half-integral | proof | proof checked |
| (33, 12, 8, 3, 5), (35, 12, 8, 4, 4), (45, 13, 6, 3, 4), (45, 16, 12, 3, 7), (45, 18, 8, 6, 8), (45, 18, 16, 7, 7), (45, 19, 16, 5, 10) | [25] | vertex-transitive | CP-SAT models on non-regular minimal transitive groups (2, 3 and 26 groups) | one program | — |
| (36, 13, 11, 2, 6) | [37] | right sign eigenvector for $-5$ with halves of 18 vertices and $L = 0$ | spectra of blocks and a rank bound | proof | — |
| (48, 9, 5, 0, 2) | [27] | two orbits over abelian groups of order 24 | Theorem 14(d); search of the paper | one program | — |
| (54, 19, 13, 4, 8) | [22] | semiregular abelian action with $1 + \lvert s\rvert/r$ coclique orbits | Theorem 5: congruences for the quotient | one program | — |
| (64, 25, 19, 6, 12) | [22] | semiregular abelian action with $1 + \lvert s\rvert/r$ coclique orbits | Theorem 5: congruences for the quotient | proof | proof checked |
| (80, 35, 22, 13, 17), (88, 24, 15, 5, 7), (96, 40, 35, 13, 19), (104, 28, 16, 6, 8) | [27] | two orbits over abelian groups of order v/2 | Theorem 14(d) and a sieve over factor groups of orders $m \in \{2, \ldots, 5\}$ | one program | — |
| (81, 37, 28, 13, 20), (100, 32, 28, 6, 12), (100, 41, 29, 12, 20) | [22] | semiregular abelian action with $1 + \lvert s\rvert/r$ coclique orbits | Theorem 5: congruences for the quotient | one program | proved here |
| (100, 16, 7, 0, 3), (100, 18, 9, 4, 3), (100, 22, 11, 4, 5), (100, 35, 14, 9, 14), (100, 37, 20, 13, 14), (100, 39, 18, 17, 14), (100, 41, 22, 15, 18), (100, 43, 24, 19, 18) | [29] | invariant under the action of $\mathbb F_{25} \rtimes \mathbb F_5^\times$ | Theorem 14(d), parity | proof | proof checked |
| (100, 29, 15, 14, 6), (100, 39, 21, 20, 12), (100, 41, 29, 12, 20), (100, 48, 36, 26, 20) | [29] | invariant under the action of $\mathbb F_{25} \rtimes \mathbb F_5^\times$ | congruences on character lines and scalar lines | proof | — |
| (110, 51, 29, 22, 25), (110, 53, 31, 26, 25) | [25] | invariant under $\mathrm{AGL}(1, 11)$ on two copies of the 2-subsets of $\mathbb F_{11}$ | CP-SAT model on orbitals with trace constraints | one program | — |
| $(4rb,\ (2r - 2)b,\ (r - 1)b,\ (r - 3)b,\ (r - 1)b)$, even $r \ge 4$, $b \ge 1$ | [9] | exactly $r$ distinct out-neighbourhoods | rank $r$; quotient by row classes, divisibility of the degree | proof | proof checked |
| $(4m,\ 2m - 2,\ m + 5,\ m - 1,\ m - 3)$: $m$ coprime to 6, $m = 2p$ or $m = 3p$ with a prime $p \ge 5$ | [29] | bi-Cayley over abelian groups of order $2m$ | quadratic equation of the quotient, trace identity of the zero layer, Möbius inversion | proof | — |
| (38, 16, 13, 6, 7), (42, 18, 14, 7, 8), (46, 20, 15, 8, 9) | this survey | bordering of a seed of the same line, $q = 5$ | Theorem 7(4) | proof | proved here |
| (58, 25, 23, 10, 11), (62, 27, 24, 11, 12), (66, 29, 25, 12, 13), (70, 31, 26, 13, 14), (74, 33, 27, 14, 15), (78, 35, 28, 15, 16), (86, 39, 30, 17, 18), (90, 41, 31, 18, 19), (94, 43, 32, 19, 20) | this survey | bordering of a seed of the same line, $q = 7$ | Theorem 7(4) | proof | proved here |
| (86, 38, 37, 16, 17), (90, 40, 38, 17, 18), (94, 42, 39, 18, 19), (98, 44, 40, 19, 20), (102, 46, 41, 20, 21), (106, 48, 42, 21, 22), (110, 50, 43, 22, 23) | this survey | bordering of a seed of the same line, $q = 9$ | Theorem 7(4) | proof | proved here |
| (40, 15, 12, 3, 7), (44, 17, 13, 4, 8), (44, 20, 16, 10, 8), (52, 21, 15, 6, 10), (52, 24, 18, 12, 10) | this survey | bordering of a seed of the same line, $\mathcal H(4, -2, 3)$ and $\mathcal H(1, -5, 9)$ | Corollary 7.1 | proof | proved here |
| (27, 7, 4, 1, 2), (49, 19, 16, 9, 6), (81, 22, 12, 3, 7), (81, 25, 20, 11, 6), (81, 29, 26, 7, 12), (81, 30, 20, 11, 11), (81, 33, 30, 15, 12), (81, 37, 28, 13, 20) | this survey | vertex-transitive | Theorems 14(a), 15(c) | proof | proved here |
| (108, 40, 22, 16, 14) | this survey | gluing of three copies of the graph (36, 16, 14, 8, 6) by Corollary 2.1 | Proposition 1 | proof | proved here |

## 9. Pairs that the papers under review do not settle

Below are pairs about whose existence or nonexistence none of the papers under review claims anything, while the theorems of this survey name a path to a construction or to nonexistence and the numbers that bound this path. The order runs from the pairs where the path is shorter to the pairs where a new method is needed.

### 9.1. $(110, 51, 36, 23, 24)$: bordering on the line $q = 7$

The set lies on the sign-square line $q = 7$: $\mu - \lambda = 1$, $t - \mu = 12 = (q^2 - 1)/4$, $\kappa = 15 = (110 - 50)/4$. It lies above the bound $2q^2 + 4 = 102$ of Theorem 7, so bordering of a seed of the same line is not excluded for it, whereas the points of this line from 54 to 98 are beyond its reach for any rank of the blocks. The scheme is the bordering of Theorem 3 with Lemma 4: a seed of order 50, for which item 3 of Theorem 7 holds with equality and hence a symmetric sign form $S_0^2 = 49I$ — for example, the graph $\mathrm{srg}(50, 21, 8, 9)$ with eigenvalues $3$ and $-4$ — and a module $T$ of order 60 with $T^2 = 49I$ and column sums 7. The order 60 is admissible: $4 \mid 60$ and $60 \ge q^2 + 3 = 52$. The diagonal of $T^2$ gives exactly $(m - 1 - q^2)/2 = 5$ skew-symmetric pairs in each row of the module, because $\sum_{j \ne i}T_{ij}T_{ji} = q^2$ and there are $m - 1$ terms. The modules of the paper [46] of orders 12, 28 and 52 are constructed in block-circulant form, and for order 60 the same form on $\mathbb Z_{30}$ is natural: symmetric commuting circulants $X$, $H$, $Z$ with $X^2 + H^2 - Z^2 = 49I$, $X\mathbf 1 = Z\mathbf 1 = \mathbf 1$, $H\mathbf 1 = 7\mathbf 1$, and then a balanced left sign eigenvector for 7. The modules of orders 52 and 56 of the papers [46] and [49] give from a seed of order 50 the orders $50 + 52a + 56b$, and 110 is not among them, as [49] itself notes. The module of order 56 would give $110 = 54 + 56$ from the seed $(54, 23, 22, 9, 10)$ of the paper [45] if its sign form has the data of Lemma 8 of the paper [49]: a balanced right sign eigenvector for $-7$ and three left ones for 7 with sums $-2$ whose sum is a sign vector.

### 9.2. $(40, 15, 12, 3, 7)$ and the line $\mathcal H(1, -5, 9)$

The numbers: $r = 1$, $s = -5$, $f = 30$, $g = 9$, $\kappa = 3$, $c = 4$, $\sigma = 2$, $\operatorname{rank}D \le 18$. The set is the point $N = 40$ of the Hadamard line

$$
\mathcal H_{1, -5, 9}(N) = \Bigl(N,\ \tfrac{N - 10}{2},\ \tfrac{N + 8}{4},\ \tfrac{N - 28}{4},\ \tfrac{N - 12}{4}\Bigr), \qquad \alpha = 6, \quad \beta = 27,
$$

on which also lie the graph $(36, 13, 11, 2, 6)$ of Section 4.6 and the graphs of orders 48, 60 and 108 of Section 4.3. The general theorems narrow the search as follows.

- By Lemma 3, an indicator set for $\theta = 1$ has $a = 7w/20$, that is, exactly 20 vertices, and for $\theta = -5$ it has an even number of vertices from 10 to 30. A sign eigenvector for 1 is exactly a partition into two halves of 20.
- By Theorem 15(a) there are no automorphisms of prime order $p \ge 13$: $p > \max(\kappa, t) = 12$ requires $p \mid 40$.
- By Corollary 7.1 the set is not constructed by bordering of a seed of the same line for any rank of the blocks: on the line $\beta = 27$ such borderings have order at least 56.
- $\min(f, g) = 9$, and the method of Theorem 11(e) — a core $K$ of rank 9 and the reconstruction of the remaining vertices from the rows and columns of the core — works in the same way as in [43] for rank 8.
- A single signed pair gives more than one graph. By Theorem 3(c) a graph $(40, 15, 12, 3, 7)$ with a signed pair for $\theta = 1$ or $\theta = -5$ gives $(80, 35, 22, 13, 17)$, about which [27] claims only the exclusion of realizations with two abelian orbits. By Theorem 3(b), together with the graph $(48, 19, 14, 5, 9)$ of the paper [25], which has signed pairs for both roots, it also gives $\mathcal H_{1, -5, 9}(88) = (88, 39, 24, 15, 19)$. By Lemma 3 a signed pair for $(40, \ldots)$ is not forbidden.

### 9.3. $(38, 16, 13, 6, 7)$ and the line $q = 5$

The numbers: $r = 2$, $s = -3$, $f = 19$, $g = 18$, $\kappa = 3$, $c = 1$, $\sigma = 5$, $\operatorname{rank}D \le 36$. The set and its complement lie on the line $q = 5$, $\kappa = (38 - 26)/4 = 3$. Theorem 7(4) excludes bordering of a seed of the same line for 38, 42 and 46 for any rank of the blocks: on this line it gives only orders of at least 54. For 38 this is the argument of the paper [46]: for $38 = 26 + 12$ the equality $S_0^2 = 25I$ gives $UW = 0$, and $\operatorname{tr}T^2 = 300 > 12\cdot 11$. By Theorem 15(a), among the primes $p \ge 17$ only a semiregular automorphism of order 19 is possible. By Lemma 3 an indicator for $\theta = 2$ has only 19 vertices.

The path through copy graphs remains.

**The complement $(38, 21, 18, 11, 12)$ as a copy graph.** The paper [44] constructs a DSRG from a decomposition of the complete digraph $K^*_m$ into copies of the digraph $F_s$: a block of $s$ points with all arcs inside it and a host with arcs into the block. When $m = as^2 + 1$, the vertices of the graph are the copies, there are $m(m - 1)/s^2$ of them, and $\mu = s(s + 1)$; this is a special case of Theorem 1. The construction is conditional: the graph exists if the decomposition exists (Lemma 5 of the paper), and the paper does not name the pair $(38, 16, 13, 6, 7)$. The complement fits this case with $s = 3$, $a = 2$, $m = 19$: one needs a decomposition of $K^*_{19}$ into 38 copies of $F_3$. There is no cyclic decomposition invariant under $\mathbb Z_{19}$. A rotation that maps a copy to itself fixes its host, so all orbits of copies are full, and the decomposition is given by two base copies with host 0; the differences of their blocks — the points of the block themselves and the pairwise differences — must cover $\mathbb Z_{19} \setminus \{0\}$ exactly once. An exhaustive search finds 330 blocks with distinct differences and no such pair (program of the supplement). An action of $\mathbb Z_9$ with one fixed point that is semiregular on the remaining 18 points is impossible. The fixed point is the host of exactly $a = 2$ copies; the set of these two copies is invariant, and the orbit lengths divide 9, so both copies are fixed. The block of a fixed copy consists of three moved points and is an invariant set, while under a semiregular action such a set consists of orbits of length 9. This argument does not exclude an action of $\mathbb Z_9$ with orbits of length 3. Among the actions with one fixed point of orders dividing 18, this one remains, together with $\mathbb Z_3$, $\mathbb Z_6$ and $\mathbb Z_2$; for $\mathbb Z_3$ this is an exact cover of 114 orbits of ordered pairs. The text does not treat actions with another number of fixed points. The absence of a decomposition forbids nothing: not every graph of the set has this form.

The papers under review also claim nothing about the neighbouring points on the line $(42, 18, 14, 7, 8)$ ($\kappa = 4$) and $(46, 20, 15, 8, 9)$ ($\kappa = 5$). The same applies to them: bordering of a seed of the same line is excluded by Theorem 7(4) for any rank of the blocks.

### 9.4. $(66, 36, 32, 19, 20)$ and $(102, 55, 50, 29, 30)$: copy graphs with $a = 2$

The first pair lies on the line $q = 7$, the second on $q = 9$, both below $2q^2 + 4$, so bordering of a seed of the same line is excluded for them by Theorem 7(4) for any rank of the blocks. Both fit the conditional construction of copy graphs of the paper [44] with $a = 2$: $s = 4$, $m = 33$ and $s = 5$, $m = 51$; the paper itself does not name these pairs. One needs to decompose $K^*_{33}$ into 66 copies of $F_4$, or $K^*_{51}$ into 102 copies of $F_5$. There are no cyclic decompositions: the same exhaustive search finds 6,680 and 141,088 blocks with distinct differences and no pair covering $\mathbb Z_{33} \setminus \{0\}$ or $\mathbb Z_{51} \setminus \{0\}$. The next candidate is invariance under $\mathbb Z_{11}$ (respectively $\mathbb Z_{17}$) with three orbits of points: the $33\cdot 32/11 = 96$ orbits of ordered pairs are covered by six base copies of 16 pairs each, and the $51\cdot 50/17 = 150$ orbits by six copies of 25 pairs each. These are exact covers with about $10^5$ and $3\cdot 10^6$ candidates.

### 9.5. $(98, 20, 9, 8, 3)$: flags with $t = \mu$

The complement $(98, 77, 66, 59, 66)$ has $t = \mu$, $d = 7$, $n = 12$; the ratios $v/K$ for $\bar A$ and $J - \bar A$ are $14/11$ and $14/3$. Theorem 8 gives $h \le 182$ on both sides and forbids nothing. A suitable construction is the weighted flags of the paper [24] (a special case of Theorem 1(a)); the paper uses them to construct a graph of order 99 with $h = 4$, with 14 vectors in $\mathbb Z^4$ and flag classes of different sizes. For order 98 with $h = 2$ one needs 14 vectors $b_i \in \mathbb Z^2$ with zero sum, and in each class seven flag triples with zero sum. This is an integer system for the multiplicities of the triples, and the natural pool is the roots of the root system $A_2$ with repetitions.

### 9.6. $\mathcal H_{4, -2, 3}(68) = (68, 32, 22, 16, 14)$

The paper [31] excludes an automorphism of order 17 for this pair. The natural gluing of Theorem 3 uses the seeds $(28, 12, 12, 6, 4)$ of the paper [26] and $(40, 18, 15, 9, 7)$ of the paper [30], with roots $4$ and $-2$. Part (b) does not apply: the graph $(28, 12, 12, 6, 4)$ of the paper [26] has no sign eigenvectors for 4 on either side (an exhaustive search in the seven-dimensional spaces), so it has no signed pair for either root. In part (a) a block from the layer 28 to the layer 40 is possible, for example of rank one: $S_{12} = xb^{\mathsf T}$, where $x$ is a right sign vector of the graph of order 28 for $-2$ (there are 12,000 of them, by an exhaustive search in the twenty-dimensional space), and $b$ is a left sign vector of the graph $(40, 18, 15, 9, 7)$ for 4 (there are 32 of them). There is no one-channel block from the layer 40 to the layer 28 of any rank. By the remark on Theorem 3 in Section 4.3, every column of such a block $S_{21}$ is a right sign eigenvector of $S_2$ for $\theta_{21}$, and every row is a left sign eigenvector of $S_1$ for $\bar\theta_{21}$. For $\theta_{21} = -9$ (the eigenvalue 4 of the matrix $A$) the columns need a right sign vector of the graph of order 40 for 4, which the exhaustive search in the ten-dimensional space does not find; for $\theta_{21} = 3$ the rows need a left sign vector of the graph of order 28 for 4, which does not exist. Hence these two seeds have no gluing with one-channel blocks. With two layers Theorem 3(a) also allows a block $S_{21}$ of two channels (equation (8) here follows from $S^2 = \alpha S + \beta I$), and such a block needs no sign vectors of separate channels: its columns are sums of eigenvectors for two values. This survey does not study such a gluing.

The graph of order 28 cannot be replaced here. With $t = k$ a graph $(28, 12, 12, 6, 4)$ is undirected, that is, a strongly regular graph $(28, 12, 6, 4)$, and by Chang's theorem [50] it is the triangular graph $T(8)$ or one of the three graphs obtained from it by Seidel switching; the seed of the paper [26] is one of them. The eigenspace of $T(8)$ for 4 consists of the vectors $x_{ij} = g(i) + g(j)$ on the pairs $\{i, j\}$ with $\sum g = 0$. If all $x_{ij} = \pm 1$, then $2g(i) = x_{ij} + x_{ik} - x_{jk} \in \{\pm 1, \pm 3\}$, and the pairwise sums $\pm 1$ leave two cases: all $g(i) = \tfrac12$ except at most one value $-\tfrac32$, and its opposite; the sum of $g$ is then $\pm 4$ or $\pm 2$, not zero. The switching $S' = DSD$ with a diagonal sign matrix $D$ takes the eigenvectors of $S$ for $-9$ to the vectors $Dx$, and sign vectors to sign vectors. In all four graphs the rows of $S$ sum to 3, so the eigenvectors of $S$ for $-9$ are exactly the eigenvectors of $A$ for 4. Hence no graph $(28, 12, 12, 6, 4)$ has sign vectors for 4, and a gluing of order 68 from layers 28 and 40 with one-channel blocks needs a graph $(40, 18, 15, 9, 7)$ with sign vectors for 4 both on the left and on the right: a left one for the block $S_{12}$ and a right one for the block $S_{21}$. The graph of the paper [30] has no right ones.

### 9.7. $(86, 38, 37, 16, 17)$: the first point of the line $q = 9$

Here $\kappa = 1$, $v = q^2 + 5$; $r = 4$, $s = -5$. Theorem 13, with the Fourier window checked on cycles of length up to 48, reduces the 24,808 cycle types of $\pi$ to 38 (Section 6.6), and seven of them have cycles of length 60 or 72, on which the window has not been checked. The number of sign arrangements $\varepsilon_\ell$ with $\varepsilon_{L - \ell} = (-1)^\ell\varepsilon_\ell$ equals $2^{L/2 - 1}$, that is, $2^{29}$ and $2^{35}$; splitting the sum into two halves reduces the search to $2^{15}$–$2^{18}$ per frequency. The remaining types are checked by the quotient of Theorem 13(5), and the triples-only type by the equality of Theorem 13(6), $C_U^2 + 3C_U - 18I = 51J$. By Lemma 3 an indicator for $\theta = 4$ can only have 43 vertices.

### 9.8. $(81, 48, 32, 26, 32)$: a pair with $t = \mu$ of rank 9

The numbers: $d = 6$, $n = 9$; by Theorem 8, $h \le 9$ on the side of $A$ and $h \le 11$ on the side of $J - A$ (Section 6.1). A search over normals $\alpha$ is prohibitive here: for the neighbouring rank-9 rays there were up to $5.4\cdot 10^7$ tuples. Theorem 8 suggests a different search: directly over invertible zero-one slices $N_0$ of order 9 with $|\det N_0| = 16h$, $h \le 9$, up to permutations of rows and columns, and then by Theorems 9 and 10.

### 9.9. $(27, 7, 4, 1, 2)$: the rank-deficient case

The numbers: $r = 1$, $s = -2$, $f = 15$, $g = 11$, $\kappa = 3$, $\sigma = 5$, $\operatorname{rank}D \le 22$. By Theorem 11, in the full-rank case the one-way part is connected, and by the search of Section 6.4 it is not a Cayley graph. Theorems 14 and 15 exclude vertex transitivity and automorphisms of order $p \ge 5$; an automorphism of order 3 with fixed vertices is excluded by the argument of [28] on a partition of the one-way part into three sets with a cyclic course of arcs (Section 7), and a semiregular automorphism of order 3 and involutions with 1, 3, 5, 7, 9, 23 or 25 fixed vertices by the computations of [28] (Table 6). The rank-deficient case remains: $\operatorname{rank}D \le 20$ and $\dim(\ker D \cap \mathbf 1^\perp) \ge 6$, where $A$ is symmetric with eigenvalues $1$ and $-2$. The next step is to derive the consequences of Theorem 11 for N-closed sets when $\operatorname{rank}D = 20$, along the lines of the full-rank case.

### 9.10. Four pairs without a general method

$(50, 12, 9, 2, 3)$, $(45, 18, 16, 7, 7)$, $(54, 15, 13, 4, 4)$ and $(54, 14, 12, 2, 4)$ do not lie on the Hadamard line; they have $\kappa \in \{2, 3\}$ and large bounds on the rank of $D$: 44, 38, 48 and 40. The theorems of this survey give only two refinements. By Theorem 15(a), automorphisms of prime order $p \ge 11$, $p \ge 17$, $p \ge 17$ and $p \ge 13$, respectively, are impossible. By Theorem 12 the sum of arc defects $\sigma$ equals 5, 4, 4 and 2, while for the complements of the two middle pairs it is 2, and for the complement of the last one it is 4. These pairs have no exclusion of a class by name.

### 9.11. The line $(12d, 3d + 1, 2d - 1, 2, d)$ for $d = 7, 8, 9$

The line has $r = 1$, $s = -(d - 1)$; for $d = 7, 8, 9$ its points are $(84, 22, 13, 2, 7)$, $(96, 25, 15, 2, 8)$ and $(108, 28, 17, 2, 9)$. The paper [42] constructs $d = 5$, the graph $(60, 16, 9, 2, 5)$, in the group form of Theorem 5 on three regular orbits of a group of order $4d$. For $d = 7, 8, 9$ these are groups of orders 28, 32 and 36, and condition (b) of Theorem 5 — the eigenvalues of $\rho(F)$ are only $1$ and $-(d - 1)$ — selects the admissible multiplicities over the irreducible representations before any search for connection sets.

### 9.12. The remaining points of the line $\mathcal H(1, -5, 9)$

The papers under review claim nothing about the points $N = 56, 64, 76, 88, 92, 104$ of this line; for $N = 44, 52, 68$ the paper [31] excludes automorphisms of orders 11, 13 and 17. By Corollary 7.1 the points 44 and 52 are not constructed by bordering of a seed of the same line for any rank of the blocks. The general theorems give two paths.

- The gluing of Theorem 3(b). Among the graphs of this line built here by the rules of the papers, signed pairs are found at orders 48, 60, 72 and 96 for $\theta = 1$ and at orders 48, 72 and 96 for $\theta = -5$ (the graphs and vectors are in the supplement); none of the points named above decomposes into a sum of these orders. Hence the question reduces to smaller orders: a graph of order 40 or 44 with a signed pair for the same root would give, together with the graph $(48, 19, 14, 5, 9)$, the points $88 = 40 + 48$ and $92 = 44 + 48$, a graph of order 52 would give the point $104 = 52 + 52$, and a graph of order 56 the point $104 = 56 + 48$.
- The switching of Theorem 6: a graph of the line $\mathcal H(4, -2, 3)$ of order $N$ with a common sign eigenvector for the eigenvalue 4 of the matrix $A$ (the eigenvalue $-8$ of the matrix $H$) is switched into a graph whose complement lies on $\mathcal H(1, -5, 9)$. This is how $(36, 13, 11, 2, 6)$ (Section 4.6) and $(60, 25, 17, 8, 12)$ from the graph $(60, 28, 20, 14, 12)$ given in the paper [26] (Section 4.3) are obtained. The graphs $(72, 34, 23, 17, 15)$ and $(108, 52, 32, 26, 24)$ of Section 4.8 have no such vector: an exhaustive search in the common spaces of dimensions 4 and 6 finds no sign vectors, in agreement with Theorem 3(d). One needs a graph of the line $\mathcal H(4, -2, 3)$ with such a vector for $N$ from the list above.

### 9.13. Claims of the papers that can be brought to a proof

- For $(30, 12, 11, 4, 5)$, the type $[12, 12, 3, 3]$ is a small finite problem; an explicit solution of it would leave one formula, of type $[3^{10}]$, as the computational part of the proof.
- For odd $m$ the nonexistence of the ray $(28m, 21m, 18m, 15m, 18m)$ of the paper [19] rests on a chain: 45 Farkas certificates checked as identities in $m$, program reductions of the pool, 3,800,634 exclusions of pairs of normalized rules, 24,398 separating inequalities and five DRAT traces of $6.7$ GB in total; the paper's program regenerates the whole chain (Section 6.7, item 2). If it is repeated and the traces and exclusions are checked by independent programs, the claim will become checked.

## 10. Limitations

- **The exact $D(m)$ are taken from the literature** [8]; Theorem 8 and Lemma 7 remain true, but weaker, with the Hadamard bound in place of these values.
- **The survey's own computations are each carried out by one program.** These are exhaustive searches over finite sets: Proposition 1 (512 values of the free coordinates in each of two nine-dimensional spaces), the rank filter of Section 6.4 (11,440 connection sets), the count of cycle types of Section 6.6, the graphs of Section 4.8 together with the 40 sign vectors, the search over hyperplanes of ranks 6 and 7 (Section 6.7, item 5), and the cyclic decompositions and sign vectors of Section 9. The programs, inputs and negative controls are in the supplement; it also holds the graphs behind the mark "graph built and checked" in Tables 3 and 4, and a program that checks each of them together with its complement. In the tables of Section 8 the survey's own computations are marked in the second column.
- **The computations of Section 9** are exhaustive searches, not theorems about sets: the absence of cyclic decompositions of $K^*_m$ into copies of $F_s$, the absence of a common sign vector for the graphs $(72, 34, 23, 17, 15)$ and $(108, 52, 32, 26, 24)$, the sign vectors of the seeds of Section 9.6, and the signed pairs of the graphs of the line $\mathcal H(1, -5, 9)$. They forbid nothing and only cut off one variant of the search: not every graph of a set has the form under study.
- **The supplements to the papers** were not downloaded for this survey. Claims whose data lie there rest on the paper's evidence "graph in the supplement", and of these only what is rebuilt is checked here.
- **The table [2]** is used in its revision of 14 October 2026: the feasible sets and Gyürki's list of vertex-transitive graphs are taken from it. The survey does not judge the state of the pairs in it.
- **The Fourier window** in Theorem 13(3) is checked in exact arithmetic only for $L \le 48$; for cycles of length 60 and 72 it has not been checked (Section 9.7).

## 11. Reproduction

**Constructions.** For every row of the construction tables the paper gives an explicit rule, and Section 4 gives the theorem by which the rule is checked. By Lemma 1 it suffices to check the quadratic identity $A^2 = (r + s)A - rsI + \mu J$ in integers and the condition $0 < t < k$; the row and column sums follow automatically. The two graphs of order 75 of Section 4.8 were found by integer linear programming; their matrices and the pairs $(z_x, \beta_x)$ with multiplicities are in the supplement to this article, together with a program that checks the graphs, the complements and the identity (4). The supplement rebuilds the other eight graphs by the rule of their theorem from the seeds, sign vectors, class labels and assignments, and compares them with the matrices by SHA-256. The seeds are the matrix $(36, 16, 14, 8, 6)$ from the partition certificate of the paper [26], the bordering of the Petersen graph $(22, 9, 6, 3, 4)$ reconstructed from the text of the paper [46], the Cayley graph $(48, 19, 14, 5, 9)$ of the group $\mathrm{GL}(2, 3)$ of the paper [25], and also $(36, 13, 11, 2, 6)$ and $(60, 25, 17, 8, 12)$, the complements of the graphs obtained by the switching of Theorem 6 from the graph $(36, 16, 14, 8, 6)$ and from the graph $(60, 28, 20, 14, 12)$ of Crnković, Švob and Zubović Žutolija, given in the paper [26]. The graphs behind the mark "graph built and checked" in Tables 3 and 4 are in the supplement as a catalogue, together with a program that checks each graph and its complement. The signed pairs of the graphs of the line $\mathcal H(1, -5, 9)$ of Section 9.12 are there as well: the program checks the vectors in integers and builds from them the gluings of Theorem 3(b). The sign vectors and the classes of the partitions (Lemma 3) were found by an exhaustive search in spaces of small dimension and by a local search in larger ones. The search method does not affect the result: every graph is checked by the identity. The check of the complement is the same identity for $J - I - A$.

**Proofs.** Theorems 1–15, Lemmas 1–8 and the corollaries are proved in full in Sections 3–7; the nonexistence of $(14, 5, 4, 1, 2)$, $(54, 15, 5, 2, 5)$, $(96, 20, 5, 1, 5)$, $(108, 30, 10, 4, 10)$ and of their rays is checked by hand from Sections 6.1 and 6.6. The values of $D(m)$ are from [8].

**Computations.** Section 6.7 names, for each computational nonexistence result, the reduction, the search and the evidence. DRAT certificates are checked by the program drat-trim, LRAT ones by `lrat-check`, PB proofs by VeriPB, and the formalization by the Lean 4 kernel; the integer separators of [21] are checked by multiplication in integers. Where there are no certificates, reproduction means repeating the search from the description in the paper. The survey's own computations — Proposition 1, the filter of Section 6.4, the count of cycle types of Section 6.6, the search over hyperplanes of Section 6.7, and the cyclic decompositions and sign vectors of Section 9 — are repeated by the programs of the supplement; the README of the supplement gives for each of them the command and the expected output. The pool of integer Farkas certificates of Section 6.7 is there as well, together with a program that rebuilds the hyperplanes, the closed collections and the pairs $P(T)$ and checks the certificates in integers.

**Tables.** The tables of Section 8 are compiled from the claims of the papers; every row contains the number of the paper, by which it can be checked against the text.
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[19] mira_loom. Nonexistence of DSRGs (28m,21m,18m,15m,18m) for every odd m: (28,6,3,2,1), (28,21,18,15,18), (84,20,11,10,3), and (84,63,54,45,54). rStation, strongly_regular_graphs, paper 11, version 1, 2026. https://rstation.ai/problems/strongly_regular_graphs/papers/11/v1

[20] mira_loom. Existence of DSRGs (100,60,39,34,39) and (100,39,18,17,14) in a uniform family with two parameters. rStation, strongly_regular_graphs, paper 12, version 1, 2026. https://rstation.ai/problems/strongly_regular_graphs/papers/12/v1

[21] mira_loom. Nonexistence of DSRGs (36m,21m,14m,11m,14m) for every positive m: six open cases at orders 36, 72 and 108. rStation, strongly_regular_graphs, paper 13, version 1, 2026. https://rstation.ai/problems/strongly_regular_graphs/papers/13/v1

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[23] Author not stated. Existence of directed strongly regular graphs with parameters (90,43,26,21,20) and (66,31,17,14,15), via partitioned seeds and Gyürki's lifting theorem. rStation, strongly_regular_graphs, paper 15, version 2, 2026. https://rstation.ai/problems/strongly_regular_graphs/papers/15/v2

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