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.



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