Results on Laplacian spectra of graphs with pockets

Let F, Hy be simple connected graphs on n and m 1 vertices, respectively. Let v be a specified vertex of H-v, and u(1), . . . , u(k) is an element of F. Then the graph G = G[F,u(1), ...,u(k), H-v] obtained by taking one copy of F and k copies of H-v, and then attaching the ith copy of H-v, to the vertex u(i), i = 1, . . . , k, at the vertex v of H-v, (identify u(i) with the vertex v of the ith copy) is called a graph with k pockets. In 2008, Batik raised the question that 'how far can the Laplacian spectrum of G be described by using the Laplacian spectra of F and H-v?' and discussed the case when deg(v) = m in H-v. In this article, we study the problem for more general cases and describe the Laplacian spectrum. As an application, we construct new nonisomorphic Laplacian cospectral graphs from the known ones. (C) 2017 Kalasalingam University. Publishing Services by Elsevier B.V.