Spectral property of certain class of graphs associated with generalized Bethe trees and transitive graphs

Yi-Zheng Fan,Li Shuang-Dong,Dong Liang

doi:10.2298/aadm0802260f

Abstract

A generalized Bethe tree is a rooted tree for which the vertices in each level having equal degree. Let Bk be a generalized Bethe tree of k level, and let T r be a connected transitive graph on r vertices. Then we obtain a graph Bk?T r from r copies of Bk and T r by appending r roots to the vertices of T r respectively. In this paper, we give a simple way to characterize the eigenvalues of the adjacency matrix A(Bk ? T r) and the Laplacian matrix L(Bk?T r) of Bk?T r by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(Bk?T r) and the Fiedler vectors of L(Bk ? T r). In addition, we obtain some results on transitive graphs.

Highlights

Let G be a simple graph on n vertices
The Laplacian matrix of G is defined to an n×n matrix L(G) = D(G)−A(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees
A generalized Bethe tree Bk of k levels is introduced by Rojo and Soto [7] and is defined to a rooted tree with the vertices in each level having equal degree

Summary

INTRODUCTION

Let G be a simple graph on n vertices. The Laplacian matrix of G is defined to an n×n matrix L(G) = D(G)−A(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees. A generalized Bethe tree Bk of k levels is introduced by Rojo and Soto [7] and is defined to a rooted tree with the vertices in each level having equal degree. They found the eigenvalues of the adjacency matrix and Laplacian matrix of Bk, which are respectively the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k whose entries are given in terms of the vertex degrees. Motivated by the work of Rojo et al, we consider more general graph Bk◦T r, which is obtained from r copies of Bk and a connected transitive graph T r on r vertices by appending r roots to the vertices of T r respectively.

PRELIMINARIES

SPECTRUM OF THE LAPLACIAN MATRIX

SPECTRUM OF THE ADJACENCY MATRIX

PROPERTY OF EXTREME EIGENVECTORS