Abstract

A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. For i=1,2,...,p, let Bi be a generalized Bethe tree of ki levels and let Δi1,2,...,ki-1 such that (1) the edges of Bi connecting vertices at consecutive levels have the same weight, and (2) for j∈Δi, each set of children of Bi at the level ki-j+1 defines a clique in which the edges have weight ui,j. For i=1,2,...,p, let Gi be the graph obtained from Bi and the cliques at the levels ki-j+1 for all j∈Δi. Let G be the graph obtained from the graphs Gi 1≤i≤p joined at their respective roots. We give a complete characterization of the eigenvalues, including their multiplicities, of the Laplacian, signless Laplacian and adjacency matrices of the graph G. Finally, we characterize the normalized Laplacian eigenvalues when G is an unweighted graph. © 2012 Elsevier Inc. All rights reserved.