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arXiv · 2404.09902

Divisible design graphs from the symplectic graph

Bart De Bruyn·Sergey Goryainov·Willem Haemers·Leonid Shalaginov
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Abstract

A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a (group) divisible design. Divisible design graphs were introduced in 2011 as a generalization of $(v,k,\lambda)$-graphs. Here we describe four new infinite families that can be obtained from the symplectic strongly regular graph $Sp(2e,q)$ ($q$ odd, $e\geq 2$) by modifying the set of edges. To achieve this we need two kinds of spreads in $PG(2e-1,q)$ with respect to the associated symplectic form: the symplectic spread consisting of totally isotropic subspaces and, when $e=2$, a special spread consisting of lines which are not totally isotropic. Existence of symplectic spreads is known, but the construction of a special spread for every odd prime power $q$ is a major result of this paper. We have included relevant back ground from finite geometry, and when $q=3,5$ and $7$ we worked out all possible special spreads.

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Bart De Bruyn, Sergey Goryainov, Willem Haemers, Leonid Shalaginov. 2024-04-15. Divisible design graphs from the symplectic graph. https://arxiv.org/abs/2404.09902

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