Godsil-McKay switching and isomorphism

TL;DR

This paper investigates conditions under which Godsil-McKay switching preserves graph isomorphism, providing new sufficient conditions and examples, and analyzing the spectral properties of certain graph products.

Contribution

It introduces a straightforward sufficient condition for isomorphism after Godsil-McKay switching and explores spectral properties of specific graph products.

Findings

A new sufficient condition for isomorphism after switching

Examples showing the condition is not necessary

The tensor product of certain grids is not determined by spectrum

Abstract

Godsil-McKay switching is an operation on graphs that doesn't change the spectrum of the adjacency matrix. Usually (but not always) the obtained graph is non-isomorphic with the original graph. We present a straightforward sufficient condition for being isomorphic after switching, and give examples which show that this condition is not necessary. For some graph products we obtain sufficient conditions for being non-isomorphic after switching. As an example we find that the tensor product of the $\ell\times m$ grid ($\ell>m\geq 2$) and a graph with at least one vertex of degree two is not determined by its adjacency spectrum.