The spectra and the signless Laplacian spectra of graphs with pockets

Let G[F, V-k, H-v] be the graph with k pockets, where F is a simple graph of order n >= 1, V-k = {v(1),...,v(k)} is a subset of the vertex set of F and H-v is a simple graph of order m >= 2, v is a specified vertex of H-v. Also let G[F, E-k, H-uv] be the graph with k edge-pockets, where F is a simple graph of order n >= 2, Ek = {e(1),...,e(k)} is a subset of the edge set of F and H-uv, is a simple graph of order m >= 3, uv is a specified edge of H-uv, such that H-uv - u is isomorphic to H-uv - v. In this paper, we obtain some results describing the signless Laplacian spectra of G[F, V-k, H-v] and G[F, E-k, H-uv] in terms of the signless Laplacian spectra of F, H-v and F, H-uv, respectively. In addition, we also give some results describing the adjacency spectrum of G[F, V-k,V- H-v] in terms of the adjacency spectra of F, H-v. Finally, as many applications of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral graphs. (C) 2017 Elsevier Inc. All rights reserved.