On the Laplacian spectra of graphs with pockets

Abstract

Let F,H v be simple connected graphs. Let v be a specified vertex of Hv and u 1, … ,u k ∈ F. Then the graph obtained by taking one copy of F and k copies of Hv , and then attaching the ith copy of Hv to the vertex u i i=1, … ,k, at the vertex v of Hv (identify ui with the vertex v of the ith copy) is called a graph with k pockets. The copies of the graph Hv 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.