Abstract

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let G≤m≤ be a connected weighted graph on m vertices. Let {B≤i≤ : 1 ≤ i ≥m} be a set of trees such that, for i = 1, 2,⋯, m, B≤i≤ is a generalized Bethe tree of k≤i≤ levels, the vertices of B≤i≤ at the level j have degree ≤i≤,-3+1for j + l for J = 1, 2,⋯k≤i≤ and the edges of B≤i≤ joining the vertices at the level j with the vertices at the level (j + 1) have weight Wi,k≤i≤-j for j = 1, 2,⋯ k≤i≤ - 1. Let (G≤m≤ {Bi : 1 ≤ i ≥m} be the graph obtained from G≤m≤ and the trees B≤1≤, B≤2≤,⋯, B≤m≤ by identifying the root vertex of B≤i≤ with the ith vertex of G≤m≤ A complete characterization is given of the eigenvalues of the Laplacian and adjacency matrices of Qm {Bi : 1 ≤ i ≤ m} together with results about their multiplicities. Finally, these results are applied to the particular case B1 = B2 = ⋯ = Bm.