The spectra of a graph obtained from copies of a generalized Bethe tree

We generalize the concept of a Bethe tree as follows: we say that an unweighted rooted tree is a generalized Bethe tree if in each level the vertices have equal degree. If B-k is a generalized Bethe tree of k levels then we characterize completely the eigenvalues of the adjacency matrix and Laplacian matrix of a graph B-k((r)) obtained from the union of r copies of B-k and the cycle C-r-connecting the r vertex roots. Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. (c) 2006 Elsevier Inc. All rights reserved.