Graph switching, 2-ranks, and graphical Hadamard matrices

TL;DR

This paper investigates how Seidel and Godsil-McKay switching affect the 2-rank of adjacency matrices in graphs derived from Hadamard matrices, revealing many graphs with diverse 2-rank properties and extending previous results.

Contribution

It introduces new insights into the 2-rank behavior under switching operations for graphs from Hadamard matrices, including the construction of graphs with unbounded 2-rank diversity.

Findings

Many graphs with increased 2-rank found via computer experiments.

Construction of strongly regular graphs with various 2-ranks.

Identification of graphs with unbounded 2-rank variation.

Abstract

We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order $4^m$. Starting with graphs from known Hadamard matrices of order $64$, we find (by computer) many Godsil-McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters $(63,32,16,16)$, $(64,36,20,20)$, and $(64,28,12,12)$ for almost all feasible 2-ranks. In addition we work out the behaviour of the 2-rank for a graph product related to the Kronecker product for Hadamard matrices, which enables us to find many graphical Hadamard matrices of order $4^m$ for which the related strongly regular graphs have an unbounded number of different 2-ranks. The paper extends results from the article 'Switched symplectic graphs and their 2-ranks' by the first and the last author.