On the spectral characterizations of ∞∞-graphs

A ∞∞-graph is a graph consisting of two cycles with just a vertex in common. We first look for some invariants for cospectral graphs, then we introduce a new method to determine the degree sequence of cospectral mates of a graph. In this paper, we prove that all ∞∞-graphs without triangles are determined by their Laplacian spectra and that all ∞∞-graphs, with one exception, are determined by their signless Laplacian spectra. For the exception we determine all graphs that are cospectral (w.r.t. signless Laplacian spectrum) to it.