The spectra of the adjacency matrix and Laplacian matrix for some balanced trees

By Oscar Rojo, Ricardo L. Soto. 2005

74 citations recorded by the source corpus.

Abstract Let T be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let d k − j +1 denotes the degree of the vertices in the level j . We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of T . They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k . The codiagonal entries for both matrices are d j - 1 , 2 ⩽ j ⩽ k - 1 , and d k , while the diagonal entries are zeros, in the case of the adjacency matrix, and d j , 1 ⩽ j ⩽ k , in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b8ac9db8225f5b0203ffdca7a85bf3f640b94b6d

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Spectra of Large Random Trees (2009, background): “We note that, although trees are rather simple graphs, the analysis of their eigenvalues is still rather challenging, and even in the case of highly symmetric deterministic trees explicit formulae for spectra have only been found recently [Roj08, RR07a, Roj07, RR07b, Roj06a, Roj06b, RS05 ].”
Locating the Eigenvalues of Trees (2011, background): “Eigenvalues of trees have been studied in [8, 9, 10, 11, 12].”
On the $A_{\alpha}$-spectra of trees (2016, methodology): “…Theorem 1, we calculate the spectra of certain Bethe trees, providing, in fact, more than is needed for the proof of Theorem 1: namely, in Section 2, we introduce generalized Bethe trees and give a reduction procedure for calculating their A α -spectra, thereby extending the main results of [10].”
An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree (2007, methodology): “In [3], we characterize completely the eigenvalues of the Laplacian matrix and of the adjacency matrix of Bk .”