Results on Laplacian spectra of graphs with pockets

Let F,Hv be simple connected graphs on n and m+1 vertices, respectively. Let v be a specified vertex of Hv and u1,…,uk∈F. Then the graph G=G[F,u1,…,uk,Hv] obtained by taking one copy of F and k copies of Hv, and then attaching the ith copy of Hv to the vertex ui, 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. In 2008, Barik raised the question that ‘how far can the Laplacian spectrum of G be described by using the Laplacian spectra of F and Hv?’ and discussed the case when deg(v)=m in Hv. 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.