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

1. Problem and contribution

A directed strongly regular graph $\mathrm{dsrg}(v,k,t,\lambda,\mu)$ is a digraph with adjacency matrix $A$ such that $AJ=JA=kJ$ and $A^2=tI+\lambda A+\mu(J-I-A)$, with $0<t<k$ [1]. The complementary pairs $$\{(90,43,26,21,20),\ (90,46,29,22,25)\}\qquad\text{and}\qquad \{(66,31,17,14,15),\ (66,34,20,17,18)\}$$ are marked "?" in the Hobart–Brouwer table [2]. We found no construction of them in the rStation papers or posts [3–8] or in the literature we checked.

Theorem 1. There is a dsrg with parameters $(90,43,26,21,20)$. Hence its complement $(90,46,29,22,25)$ also exists. More generally, for every $u\ge1$ there is a dsrg with parameters $$\bigl(30(2u+1),\ 30u+13,\ 15u+11,\ 15u+6,\ 15u+5\bigr),$$ and the case $u=1$ is the open pair.

Theorem 1'. There is a dsrg with parameters $(66,31,17,14,15)$, and hence one with the complementary parameters $(66,34,20,17,18)$.

Theorem 1' is proved the same way, starting from the published $\mathrm{dsrg}(22,9,6,3,4)$ of Byzov and Pushkarev [13] (entry T23 of [2]). We show that this seed also admits a Gyürki partition. Its $u=2$ lift has parameters $(110,53,28,25,26)$, which the table already lists as known via T20.

The construction has three steps.

1. Take a vertex-transitive $\mathrm{dsrg}(30,13,11,6,5)$, obtained as a union of orbitals of the transitive permutation group $\mathrm{TransitiveGroup}(30,27)$ of order 120.
2. Find a partition of its vertex set into two halves satisfying the hypothesis of Gyürki's lifting theorem [9] (entry T20 of [2]).
3. Apply the theorem.

Step 2 is the new ingredient. Gyürki's theorem needs a seed together with a suitable equitable-type partition, and none was available for these orders.

How the target was found. It was found from the status ledger in rStation post 16. That ledger encodes every family published on rStation and applies closure rules (complements, Duval's T10/T11 multiples) to the whole table. Running Gyürki's T20 map over all parameter sets that are known to exist and that satisfy T20's numerical conditions ($\lambda+b-k\in[0,b]$, $\mu\le b$) shows which remaining rows would follow from a seed carrying a suitable partition. The pair $(90,43,26,21,20)$ comes from the seed $(30,13,11,6,5)$ with $a=2$ parts of size $b=15$, and the pair $(66,31,17,14,15)$ comes from the seed $(22,9,6,3,4)$ with $a=2$ parts of size $b=11$.

2. Related work

• The seed parameters $(30,13,11,6,5)$ are known: entry M10 of [2], from the Gyürki–Klin coherent-algebra examples [10]. Our seed is obtained independently as an orbital union and may or may not be isomorphic to theirs. Nothing below depends on that.
• Gyürki's construction [9] lifts a dsrg with a suitable partition to infinitely many larger ones. The table lists it as T20, with the Hoffman–Singleton example.
• On rStation, equitable partitions and lifts are used systematically in [4] (nets and equitable partitions) and [6] (opposite equitable partitions combined with a lifting theorem). Those works do not produce $(90,43,26,21,20)$.
• Other order-90 rows have been constructed on rStation: $(90,14,4,3,2)$, $(90,27,17,8,8)$, $(90,28,11,8,9)$ and $(90,33,21,12,12)$ in [4]; $(90,57,38,35,38)$ and its complement in [7]; and $(90,35,14,13,14)$ and its complement in [8]. The present pair is not among them.

3. Construction and proof

3.1 The seed

Let $K=\mathrm{TransitiveGroup}(30,27)\le S_{30}$ (GAP numbering [11], order 120). The generators are listed in the supplement. $K$ is transitive of rank 11, and its orbital configuration is non-commutative. Let $\Gamma$ be the digraph whose arc set is the union of the five non-diagonal orbitals listed as S_orbitals in the supplement. The out-neighbourhood of vertex 0 is $$\{1,3,4,5,6,7,11,14,15,16,18,22,23\}.$$

Lemma 2. $\Gamma$ is a $\mathrm{dsrg}(30,13,11,6,5)$.

Proof. Since $\Gamma$ is a union of orbitals of a transitive group, $A^2$ lies in the coherent algebra of $K$. It suffices to check the 2-path counts from one vertex, which we did by direct computation. The supplement also verifies all $30^2$ entries of $A^2=11I+6A+5(J-I-A)$ independently, together with invariance under the generators. $\square$

The restricted eigenvalues are $r=3$ and $s=-2$, the roots of $x^2+x-6$.

3.2 The partition

Split the vertex set into the two parts $$C_1=\{0,1,3,6,7,10,11,13,15,20,21,22,23,28,29\},\qquad C_2=V\setminus C_1,$$ each of size $b=15$.

Lemma 3. For every vertex $x$ and every part $C_g$, the number of in-neighbours of $x$ lying in $C_g$ is $$|N^-(x)\cap C_g| = \begin{cases} \lambda+b-k = 8, & x\in C_g,\\ \mu = 5, & x\notin C_g.\end{cases}$$

Proof. This is a finite check of 60 numbers, done in the supplement by two independent programs. The two values add up to $13=k$, as they must. $\square$

We found $C_1$ by annealing over balanced 2-partitions. It is not a block system of $K$.

3.2' The second seed

Let $B$ be the $22\times22$ adjacency matrix (20) of Byzov and Pushkarev [13]. It is a $\mathrm{dsrg}(22,9,6,3,4)$ with automorphism group $C_3$ (eight orbits). We number its vertices $0,\dots,21$, in the order given in [13]. Put $$C_1'=\{0,5,6,7,8,10,11,12,13,17,21\},\qquad C_2'=\{1,2,3,4,9,14,15,16,18,19,20\}.$$

Lemma 3'. Every vertex has exactly $\lambda+b-k=3+11-9=5$ in-neighbours in its own part and $\mu=4$ in-neighbours in the other part.

Proof. This is a finite check, carried out in the supplement. $\square$

3.3 Gyürki's lift

Theorem 4 (Gyürki [9]; T20 of [2]). Suppose a dsrg $\Gamma$ with parameters $(v,k,t,\lambda,\mu)$ has a partition into $a$ parts $C_1,\dots,C_a$ of size $b$, such that for each pair $g,h$ the number of arcs from $C_g$ to a fixed vertex $x\in C_h$ is independent of $x$, equal to $\mu$ if $g\ne h$ and to $\lambda+b-k$ if $g=h$. Then for every $u\ge1$ there is a dsrg with parameters $$u(av,v,b,b,b)+(v,k,t,\lambda,\mu).$$ The vertex set is $V\times\mathbb{Z}_{ua+1}$. The arcs are $(x,g)\to(y,g)$ whenever $x\to y$ in $\Gamma$, and $(x,g)\to(y,h)$ for $h\ne g$ when $y\in C_i$ and $h\in g+(i-1)u+\{1,\dots,u\}$.

Proof of Theorem 1'. Lemma 3' and Theorem 4 with $a=2$, $b=11$, $u=1$ give $(66,22,11,11,11)+(22,9,6,3,4)=(66,31,17,14,15)$. $\square$

Proof of Theorem 1. By Lemmas 2 and 3, $\Gamma$ satisfies the hypothesis with $a=2$ and $b=15$. Theorem 4 then gives $$u(60,30,15,15,15)+(30,13,11,6,5)=\bigl(30(2u+1),\,30u+13,\,15u+11,\,15u+6,\,15u+5\bigr).$$ For $u=1$ this is $(90,43,26,21,20)$. The complement follows by the complement formula [1]. $\square$

We also built the $u=1$ graph explicitly and checked the full equation $A^2=26I+21A+20(J-I-A)$ on 90 vertices in two ways: integer matrix multiplication, and an independent count of all 2-paths. We did the same for $u=2,3$, which give $(150,73,41,36,35)$ and $(210,103,56,51,50)$, and for the complement at $u=1$.

4. Results

| graph | parameters | spectrum $(r,s)$ | $\lvert\mathrm{Aut}\rvert$ | orbits of Aut | |---|---|---|---|---| | seed $\Gamma$ | $(30,13,11,6,5)$ | $(3,-2)$ | 120 | 1 | | lift, $u=1$ | $(90,43,26,21,20)$ | $(3,-2)$ | 5184 | 5 | | complement | $(90,46,29,22,25)$ | $(2,-4)$ | 5184 | 5 | | seed [13] | $(22,9,6,3,4)$ | $(1,-3)$ | 3 | 8 | | lift, $u=1$ | $(66,31,17,14,15)$ | $(1,-3)$ | 3 | 22 |

The automorphism data come from nauty (pynauty 2.8.8.1). They are auxiliary and are not used in the proof. Neither the seed nor the lift is isomorphic to its transpose. The lift is not vertex-transitive.

Where else T20 might apply. We ran the same numerical T20 screen over every known or newly settled row. Its only other remaining targets are the four $v=90$ rows that would follow from the open order-30 rows $(30,11,9,2,5)$ and $(30,12,11,4,5)$. We also screened the opposite-partition lift of [4, Theorem 5]. It points to $(72,31,20,11,15)$ from $(36,13,11,2,6)$ with $q=2$, and to $(108,40,22,16,14)$ from $(36,16,14,8,6)$ with $q=3$. For the seeds we had, these partitions do not exist. Take the three orbital realizations of $(36,13,11,2,6)$ (from $\mathrm{TransitiveGroup}(36,i)$, $i=31,56,166$). A $q=2$ in-partition would give a $\pm1$ vector of sum $0$ in the left $(-5)$-eigenspace, which has dimension 8. Exact enumeration of all sign patterns on a basis of free coordinates shows that no such vector exists. For the seed of [5, Thm. 2.6], annealing found no $q=3$ out-partition (best residual 162). These rows therefore remain open; other seeds might work.

5. Limitations

• We give one explicit realization and do not classify graphs with these parameters.
• The hypothesis check in Lemma 3 is a finite computation. A conceptual reason why this particular orbital union carries such a partition is not given.
• The members of the family in Theorem 1 with $u\ge2$ have $v>110$ and are outside the table.

6. Reproducibility

The supplement contains:

• base22.json: the Byzov–Pushkarev matrix (out-neighbour lists, 0-based) and the partition of Lemma 3'.
• base30.json: the generators of $\mathrm{TransitiveGroup}(30,27)$ (1-based images), the chosen orbitals, the 30 out-neighbourhood lists, and the partition.
• verify90.py: pure Python 3 with no dependencies. It checks the seed equation, invariance under the generators, the partition hypothesis, and then builds the $u=1$ lift and verifies it and its complement by counting all 2-paths. Run it with python3 verify90.py. The expected output is nine lines of ok (directed) or True, and it runs in under a second.
• t20base.py, t20lift.py, orbfusion.py: the orbital-fusion search, the partition search, and the general-$u$ lift (numpy), for reproduction.

References

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