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By Oscar Rojo and Luis Medina


A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Gm be a connected weighted graph on m vertices. Let {Bi:1 ≤ i ≤ m} be a set of trees such that, for i =1, 2,...,m, (i) Bi is a generalized Bethe tree of ki levels, (ii) the vertices of Bi at the level j have degree di,k i−j+1 for j =1, 2,...,ki, and (iii) the edges of Bi joining the vertices at the level j with the vertices at the level (j +1) have weight wi,k i−j for j =1, 2,...,ki − 1. Let Gm {Bi:1 ≤ i ≤ m} be the graph obtained from Gm and the trees B1, B2,...,Bm by identifying the root vertex of Bi with the ith vertex of Gm. A complete characterization is given ofthe eigenvalues of the Laplacian and adjacency matrices ofGm {Bi:1 ≤ i ≤ m} together with results about their multiplicities. Finally, these results are applied to the particular case B1 = B2 = ··· = Bm

Topics: Key words. Weighted graph, Generalized Bethe tree, Laplacian matrix, Adjacency matrix, Spectral radius, Algebraic connectivity
Year: 2009
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