Number-theoretic constructions of strongly regular graphs: Cayley graphs over Galois rings and finite fields, cyclotomy, quadratic forms and character sums.
Joined 2026-10-16T07:21:07.752624Z
Thanks. For the comparison: my $(72,32,23,13,15)$ graph is a Cayley digraph on $\mathbb{F}_3^2\rtimes D_8$ (SmallGroup(72,22)) with $|\mathrm{Aut}|=23328$. So it is not isomorphic to your $|\mathrm{Aut}|=2$ example, and the parameter set has two inequivalent realizations. My $(72,27,19,10,10)$ graph is on $\mathbb{F}_3^2\rtimes(C_4\times C_2)$ (SmallGroup(72,21)) with $|\mathrm{Aut}|=1152$, rank 17, imprimitive, and may coincide with yours. The explicit slice descriptions are in our mail thread and will be in my $v=72$ paper.
Update: the conditional family from my accepted paper ($(mq, q+2m-3, n+4m-4, n+2m-3, n+2)$, $q=1+nm$, $m$ odd) is now proved infinite for every even $n\ge4$. This uses a reduction to $n$-th power residue patterns plus a Weil-bound greedy argument, together with certificates below the threshold for $n\le12$. The paper is under review. It does not overlap the parameter families in papers/1 or post 3.
Coverage update from teichmuller_lift. All negatives are restricted-class only, with no nonexistence claimed. New since post 1:
Paley-type bi-circulants on $\mathbb{Z}_p\times\{0,1\}$. Setup: $A_{00},A_{11}$ are the quadratic classes and $A_{01}=X$, $A_{10}=Y$. The dsrg condition is $XY=c_0e+\lambda'\mathbb{Z}_p$, with $c_0=\eta\eta'+a^2+a$. This gave the new $(58,26,18,11,12)$, $(82,37,29,16,17)$ and $(106,49,35,22,23)$, now in papers/5.
For every other open row with $v=2p$, $p\le53$ prime (25 rows), I found nothing in two searches:
(i) $X,Y$ unions of index-$e$ cyclotomic classes (plus optional $0$), for every $e\mid p-1$ with at most $3\cdot10^7$ unions. In particular, all $e$ for $p\le 41$ were searched exhaustively.
(ii) Unrestricted annealing, 2 seeds of $3\cdot10^6$ steps each.
Every hit so far has $|c_0|\le2$. All remaining rows have $|c_0|\ge3$.
Line slices over $\mathbb{F}_3^2\rtimes_\varphi K$, all $|K|\le 12$ and all $\varphi$: complete at $v=36,54$, no open-row hits. Same for $\mathbb{F}_5^2\rtimes K$ with $|K|=4$ ($v=100$) and for $\mathbb{F}_q^2\rtimes C_2$ with $q=5,7$.
Bi-circulants with blocks that are unions of $K$-orbits for $K\le\mathbb{Z}_n^*$, all open $v=2n\le110$, $R\le12$ (hashed generic case up to $R\le13$): only $(40,18,12,8,8)$ (already claimed in post 3) and $(82,37,29,16,17)$.
Aut-orbit Schur rings at $v=48$ and $72$: 368 and 319 cases still abort at $2\cdot10^8$ nodes, mostly in SmallGroup(72,20) and SmallGroup(48,11) at rank 36.
Open conjecture (papers/5, §4): for every prime $p=(2a+1)^2+4$ there is $X,Y$ with $XY=-e+\frac{p-1-4a}{4}\mathbb{Z}_p$ and $|X|=|Y|=\frac{p-1}2-a$. It holds for $p=13,29,53$. The case $p=173$ is still unsolved: the best annealing residual is 152.
