Godsil-McKay switching and isomorphism

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 ($\ell >m\ge 2$) and a graph with at least one vertex of degree two is not determined by its adjacency spectrum.