On the Laplacian spectra of graphs with pockets

Let F, H(v) be simple connected graphs. Let v be a specified vertex of H(v) and u(1),...,u(k) is an element of F. Then the graph [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. The copies of the graph H(v) that are attached to the vertices u(i), i=1,...,k, are referred to as pockets. In this article, we prove some results describing the Laplacian spectrum of G using the Laplacian spectra of F and H(v). The complete Laplacian spectrum of G is also described in some particular cases.