Eigenvalues of certain weighted graphs joined at their roots having cliques at some levels

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 Δi⊆{1,2,…,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.