On the spectra of some weighted rooted trees and applications

Let T be a weighted rooted tree of k levels such that(1)the vertices in level j have a degree equal to dk−j+1 for j = 1, 2, … , k, and(2)the edges joining the vertices in level j with the vertices in level (j + 1) have a weight equal to wk−j for j = 1, 2, … , k−1.We give a complete characterization of the eigenvalues of the Laplacian matrix and adjacency matrix of T. They are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. Moreover, we give some results concerning their multiplicities. By application of the above mentioned results, we derive upper bounds on the largest eigenvalue of any weighted tree and the spectra of some weighted Bethe trees.