For i = 1, 2, ..., p, let G(i) be the graph obtained from B-i and the cliques at the levels k(i) - j + 1 for all j is an element of Delta(i). Let G be the graph obtained from the graphs G(i) (1 <= i <= p) joined at their respective roots. We give a complete characterization of the eigenvalues, including their multiplicities, of the Laplacian, signless Laplacian and adjacency matrices of the graph G. Finally, we characterize the normalized Laplacian eigenvalues when G is an unweighted graph. (C) 2012 Elsevier Inc. All rights reserved.