Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications

TL;DR

This paper analytically determines the eigenvalues of normalized Laplacian matrices for fractal trees and dendrimers, revealing their spectral properties and applications to random walks and network structure.

Contribution

It provides explicit analytical methods to compute all eigenvalues for these complex networks, advancing understanding of their spectral and dynamical characteristics.

Findings

Eigenvalues of fractal trees obtained via spectral decimation.

Eigenvalues of Cayley trees derived from recursive polynomial roots.

Explicit solutions for eigentime identity in random walks on these networks.

Abstract

The eigenvalues of the normalized Laplacian matrix of a network plays an important role in its structural and dynamical aspects associated with the network. In this paper, we study the spectra and their applications of normalized Laplacian matrices of a family of fractal trees and dendrimers modeled by Cayley trees, both of which are built in an iterative way. For the fractal trees, we apply the spectral decimation approach to determine analytically all the eigenvalues and their corresponding multiplicities, with the eigenvalues provided by a recursive relation governing the eigenvalues of networks at two successive generations. For Cayley trees, we show that all their eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. By using the relation between normalized Laplacian spectra and eigentime identity, we derive the explicit solution to the eigentime identity for random walks on the two treelike networks, the leading scalings of which follow quite different behaviors. In addition, we corroborate the obtained eigenvalues and their degeneracies through the link between them and the number of spanning trees.