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.