Wang-Qiu-Hu switching and isomorphism

Abstract

Cospectral graphs (graphs that share the same eigenvalues) expose the limitations of using the graph spectrum to uniquely identify graphs, and they also help to understand what structural properties a graph spectrum cannot capture. Switching methods, which are standard tools for constructing cospectral graphs, require specific structural and algebraic conditions to hold for the operation to preserve the graph's spectrum. However, there is no guarantee that the obtained cospectral switched graph is non-isomorphic. In this paper we study this isomorphism problem for a recent and prolific switching method to produce cospectral graphs with respect to the adjacency spectra: Wang-Qiu-Hu (WQH) switching. We do so by using common-neighbour multisets associated with a WQH partition, which allows us to derive an external common-neighbour criterion for certifying non-isomorphism after WQH-switching. Then, we apply the new criterion to clique extensions and to weak tensor products, with coclique extensions as a special case. As an application we obtain infinite families of cospectral non-isomorphic graphs, including some known constructions. Finally we extend the conditions of WQH-switching to generalized adjacency matrices and, under an additional degree condition, to Laplacian and signless Laplacian matrices.