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 B-k 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 B-k circle T-r from r copies of B-k 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(B-k circle T-r) and the Laplacian matrix L(B-k circle T-r) of B(k)oT(r) by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(B-k circle T-r) and the FIEDLER vectors of L(B-k circle T-r). In addition, we obtain some results on transitive graphs.