Explicit construction of infinite families of strongly regular digraphs with parameters $((v+(2^{n+1}-4)t)2^{n-1}, k+(2^n-2)t, t, λ, t)$
Viktor A. Byzov, Igor A. Pushkarev
TL;DR
The paper tackles the problem of generating infinite families of strongly regular digraphs by introducing a recurrence-based construction that preserves the parameter $t$ while expanding the vertex count and degree. It starts from an initial dsrg $(v,k,t,\lambda,t)$ and seeks a second graph $G_2$ whose adjacency matrix $A_2$ conforms to a structured block form relative to $A_1$, guided by a system of matrix equations and additional blockiness constraints. When such $A_2$ exists, an explicit recurrence produces a sequence of dsrg with parameters $((v+(2^{n+1}-4)t)2^{n-1}, k+(2^n-2)t, t, \lambda, t)$ for all $n\ge 1$, with a proof outline based on Kronecker-structured matrices and the auxiliary matrices $P_n$. Computational experiments using Julia and Artelys Knitro yield 11 families, including several previously open cases (e.g., $dsrg(72,18,5,3,5)$, $dsrg(76,19,5,4,5)$, $dsrg(92,23,6,5,6)$, $dsrg(104,26,7,5,7)$), demonstrating the method’s potential and highlighting future work to characterize eligible initial graphs and to derive $A_2$ without optimization-based search.
Abstract
An explicit construction of infinite sequences of strongly regular digraphs with parameter sets $((v+(2^{n+1}-4)t)2^{n-1}, k+(2^n-2)t, t, λ, t)$ is described. A computer program was used to find the initial digraphs. The remaining terms of the sequence are obtained by the constructed recurrence. Using the described approach, 11 families of strongly regular digraphs were found. In particular, these families contain digraphs $\text{dsrg}(72, 18, 5, 3, 5)$, $\text{dsrg}(76, 19, 5, 4, 5)$, $\text{dsrg}(92, 23, 6, 5, 6)$ and $\text{dsrg}(104, 26, 7, 5, 7)$, the question of the existence of which was previously open.