Switched symplectic graphs and their 2-ranks

We apply Godsil–McKay switching to the symplectic graphs over F₂ 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 ^2ν- 1 , 2 ^2ν-1, 2 ^2ν-2, 2 ^2ν-2) and 2-rank 2 ν+ 2 when ν≥ 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 ν= 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 ν≥ 3.