Divisible design graphs from the symplectic graph

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, \(e\ge 2\)) 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 that consists of lines which are not totally isotropic and which is closed under the action of the associated symplectic polarity. Existence of symplectic spreads is known, but the construction of a special spread for every odd prime power q is a main result of this paper. We also show an equivalence between special spreads of Sp(4, q) and certain nice point sets in the projective space \(\operatorname {PG}(4,q)\). We have included relevant background from finite geometry, and when \(q=3,5\) and 7 we worked out all possible special spreads.