Abstract

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Pm be a path of m vertices. Let {Bi : 1 ≤ i ≤ m} be a set of generalized Bethe trees. Let Pm {Bi : 1 ≤ i ≤ m} be the tree obtained from Pm and the trees B1, B2, ..., Bm by identifying the root vertex of Bi with the i - th vertex of Pm. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of Pm {Bi : 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 B1 = B2 = ... = Bm. © 2008 Elsevier Inc. All rights reserved.