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

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