Godsil-McKay switching and isomorphism,

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 sucient condition for being isomorphic after switching, and give examples which show that this condition is not necessary. For some graph products we obtain sucient conditions for being non-isomorphic after switching. As an example we nd that the tensor product of the grid L(';m) (' > m 2) and a graph with at least one vertex of degree two is not determined by its adjacency spectrum.