Some results on the 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 i-th copy of Hv to the vertex ui,i=1,…,k, at the vertex v of Hv (identify ui with the vertex v of the i-th copy) is called a graph with k pockets. In [Barik, S., On the Laplacian spectra of graphs with pockets, Linear and Multilinear Algebra, 56 (2008), 481-490], 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.