The spectral characterization of ∞-graphs

A ∞-graph B(r,s) is a graph consisting of two cycles Cr+1 and Cs+1 with just a vertex in common. Fan and Luo in Theorem 4.4 [8], alleged that a ∞-graph is determined by its adjacency spectrum if it contains no cycle C4. However, according to Theorem 6.3 [16], we find that the result in Theorem 4.4 [8] is not completely correct. In this paper, we prove that B(r,s) is DAS if and only if s≠=r+2, and B(r,s) has a unique cospectral mate if s=r+2.