Switched symplectic graphs and their 2-ranks

TL;DR

This paper explores how Godsil-McKay switching affects the 2-rank of symplectic graphs over GF(2), producing new strongly regular graphs with specific parameters and varying 2-ranks, expanding the known graph constructions.

Contribution

It demonstrates that switching increases the 2-rank of symplectic graphs and introduces a recursive method to generate numerous new strongly regular graphs with distinct 2-ranks.

Findings

Switching increases the 2-rank of symplectic graphs.

Many new strongly regular graphs with specific parameters were found.

A recursive construction method yields graphs with different 2-ranks for all ν ≥ 3.

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 $\nu\geq 3$. 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 $\nu\geq 3$.