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 $$\nu \ge 3$$ .