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On the spectra of some weighted rooted trees and applications

✍ Scribed by Oscar Rojo; María Robbiano

Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
221 KB
Volume
420
Category
Article
ISSN
0024-3795

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✦ Synopsis

Let T be a weighted rooted tree of k levels such that (1) the vertices in level j have a degree equal to d k-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 w k-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.

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