Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks

TL;DR

This paper derives recursive formulas to analytically determine the Laplacian eigenvalues and eigenvectors of a specific class of growing treelike networks, aiding the understanding of their spectral properties.

Contribution

It introduces a recursive analytical method to compute Laplacian spectra and eigenvectors for a class of deterministic treelike networks, expanding spectral analysis tools.

Findings

Complete eigenvalues and eigenvectors can be obtained for arbitrary network size.

The method applies recurrence relations based on network structure.

Facilitates spectral analysis of other deterministic networks.

Abstract

The complete knowledge of Laplacian eigenvalues and eigenvectors of complex networks plays an outstanding role in understanding various dynamical processes running on them; however, determining analytically Laplacian eigenvalues and eigenvectors is a theoretical challenge. In this paper, we study the Laplacian spectra and their corresponding eigenvectors of a class of deterministically growing treelike networks. The two interesting quantities are determined through the recurrence relations derived from the structure of the networks. Beginning from the rigorous relations one can obtain the complete eigenvalues and eigenvectors for the networks of arbitrary size. The analytical method opens the way to analytically compute the eigenvalues and eigenvectors of some other deterministic networks, making it possible to accurately calculate their spectral characteristics.