On the spectral characterizations of graphs

Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let p2q${\cal U}_p^{2q}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas p2q${\cal V}_p^{2q}$ the subset of p2q${\cal U}_p^{2q}$ with odd p and its q vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in p2q${\cal U}_p^{2q}$, p even and its q vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if G is a graph without isolated vertices and adjacency cospectral with the graph in pp−1={H}${\cal V}_p^{p - 1} = \{ H\}$, then G ≅ H