Explicit construction of the eigenvectors and eigenvalues of the graph Laplacian on the Cayley tree

TL;DR

This paper provides an explicit recursive construction of eigenvectors and eigenvalues of the graph Laplacian on Cayley trees, revealing spectral properties crucial for spectral analysis of non-periodic networks.

Contribution

It introduces a method to analytically derive eigenvectors and eigenvalues of the Laplacian on Cayley trees using their symmetries, advancing spectral analysis of complex networks.

Findings

Spectral gap decays exponentially with tree size.

Eigenvectors form exponentially growing eigenspaces.

Recursion relations link eigenvalues and eigenvectors.

Abstract

A generalized Fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph Laplacian. Using the symmetries of the Cayley tree, we recursively construct the family of eigenvectors with exponentially growing eigenspaces, associated with eigenvalues in the lower part of the spectrum. The spectral gap decays exponentially with the tree size, for large trees. The eigenvalues and eigenvectors obey recursion relations which arise from the nested geometry of the tree. Such analytical solutions for the eigenvectors of non-periodic networks are needed to provide a firm basis for the spectral renormalization group which we have proposed earlier [A. Tuncer and A. Erzan, Phys. Rev. E {\bf 92}, 022106 (2015)]. PACS Nos. 02.10.Ox Combinatorics; graph theory, 02.10.Ud Linear algebra, 02.30 Nw Fourier analysis