Abstract

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let {Bi : 1 ≤ i ≤ m} be a set of trees such that, for i = 1, 2, ..., m,(1)Bi is a generalized Bethe tree of ki levels,(2)the vertices of Bi at the level j have degree di, ki - j + 1 for j = 1, 2, ..., ki, and(3)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, ..., ki - 1.Let v {Bi : 1 ≤ i ≤ m} be the tree obtained from the union of the trees Bi joined at their respective root vertices. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of v {Bi : 1 ≤ i ≤ m}. Moreover, we derive results concerning their multiplicities. In particular, we characterize the spectral radii, the algebraic conectivity and the second largest Laplacian eigenvalue. © 2008 Elsevier Inc. All rights reserved.