Spectra of generalized Bethe trees attached to a path

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let P-m be a path of m vertices. Let {B-i : 1 <= i <= m} be a set of generalized Bethe trees. Let P-m {B-i : 1 <= i <= m} be the tree obtained from P-m and the trees B-1, B-2, ..., B-m by identifying the root vertex of B-i with the ith vertex of P-m. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of P-m{B-i : 1 <= i <= m}. In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B-1 = B-2 = ... = B-m. (C) 2008 Elsevier Inc. All rights reserved.