Graph switching, 2-ranks, and graphical Hadamard matrices
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 . Starting with graphs from known Hadamard matrices of order , we find (by computer) many Godsil-McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters , , and 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 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.
Keywords
Cite
@article{arxiv.1801.01149,
title = {Graph switching, 2-ranks, and graphical Hadamard matrices},
author = {Aida Abiad and Steve Butler and Willem H. Haemers},
journal= {arXiv preprint arXiv:1801.01149},
year = {2018}
}