Spectra of weighted compound graphs of generalized Bethe trees

Abstract

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Gm be a connected weighted graph on m vertices. Let {Bi :1≤ i ≤ m} be a set of trees such that, for i =1 ,2,...,m , (i) Bi is a generalized Bethe tree of ki levels, (ii) the vertices of Bi at the level j have degree di,ki−j+1 for j =1 ,2,...,ki, and (iii) the edges of Bi joining the vertices at the level j with the vertices at the level (j + 1) have weight wi,ki−j for j =1 ,2,...,k i −1. Let Gm{Bi :1≤ i ≤ m} be the graph obtained fromGm and the trees B1,B2,...,Bm by identifying the root vertex of Bi with the ith vertex of Gm. A complete characterization is given of the eigenvalues of the Laplacian and adjacency matrices of Gm {Bi :1≤ i ≤ m} together with results about their multiplicities. Finally, these results are applied to the particular case B1 = B2 = ···= Bm.