The normalized Laplacian spectrum and eigentime identities of hype-cubes

TL;DR

This paper analyzes the normalized Laplacian spectrum of hype-cubes, deriving explicit eigenvalues, eigentime identities, and spanning tree counts, revealing their structural properties and random walk behaviors.

Contribution

It explicitly determines all eigenvalues of hype-cubes' normalized Laplacian and derives formulas for eigentime identities and spanning trees, advancing spectral graph analysis.

Findings

Eigenvalues of hype-cubes are explicitly determined

Eigentime identity grows linearly with network size

Number of spanning trees is computed

Abstract

Many popular graph metrics encode average properties of individual network elements. Complementing these conventional graph metrics, the eigenvalue spectrum of the normalized Laplacian describes a network's structure directly at a systems level, without referring to individual nodes or connections. In this paper, we study the spectrum and their applications of normalized Laplacian matrices of hype-cubes, a special kind of Cayley graphs. We determine explicitly all the eigenvalues and their corresponding multiplicities by a recursive method. By using the relation between normalized Laplacian spectrum and eigentime identity, we derive the explicit formula to the eigentime identity for random walks on the hype-cubes and show that it grows linearly with the network size. Moreover, we compute the number of spanning trees of the hype-cubes.