The spectra and the signless Laplacian spectra of graphs with pockets

Let G[F, Vk, Hv] be the graph with k pockets, where F is a simple graph of order n ≥ 1, Vk={v1,…,vk} is a subset of the vertex set of F and Hv is a simple graph of order m ≥ 2, v is a specified vertex of Hv. Also let G[F, Ek, Huv] be the graph with k edge-pockets, where F is a simple graph of order n ≥ 2, Ek={e1,…,ek} is a subset of the edge set of F and Huv is a simple graph of order m ≥ 3, uv is a specified edge of Huv such that Huv−u is isomorphic to Huv−v. In this paper, we obtain some results describing the signless Laplacian spectra of G[F, Vk, Hv] and G[F, Ek, Huv] in terms of the signless Laplacian spectra of F, Hv and F, Huv, respectively. In addition, we also give some results describing the adjacency spectrum of G[F, Vk, Hv] in terms of the adjacency spectra of F, Hv. Finally, as many applications of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral graphs.