Godsil-McKay switching; Spectral characterization; Cospectral graphs; Graph isomorphism; Graph products.
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 grid L(ℓ,m) (ℓ > m>2) and a graph with at least one vertex of degree two is not determined by its adjacency spectrum.
Abiad, Aida; Brouwer, Andries E.; and Haemers, Willem H..
"Godsil-McKay switching and isomorphism",
Electronic Journal of Linear Algebra,