SPECTRAL PROPERTY OF CERTAIN CLASS OF GRAPHS ASSOCIATED WITH GENERALIZED BETHE TREES AND TRANSITIVE GRAPHS

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 Tr be a connected transitive graph on r vertices. Then we obtain a graph Bk ο Tr 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 characterize the eigenvalues of the adjacency matrix A(Bk o Tr) and the Laplacian matrix L(Bk o Tr) of Bk o Tr by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(Bk o Tr) and the Fiedler vectors of L(Bk o Tr). In addition, we obtain some results on transitive graphs.