Switched symplectic graphs and their 2-ranks

Abstract

We apply Godsil-McKay switching to the symplectic graphs over ${\mathbb{F}}_2$ with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters $(2^{2\nu }-1,2^{2\nu -1},2^{2\nu -2},2^{2\nu -2})$ and 2-rank $2\nu +2$ when $\nu \ge 3$. For the symplectic graph on $63$ vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for $\nu =3$ with various 2-ranks. Using these results and a recursive construction method for the symplectic graph from Hadamard matrices, we obtain several graphs with the above parameters, but different 2-ranks for every $\nu \ge 3$.