Spectral properties of hypercubes with applications

Abstract

In this paper, we study spectral properties of the hypercubes, a special kind of Cayley graphs. We determine explicitly all the eigenvalues and their corresponding multiplicities of the normalized Laplacian matrix of the hypercubes by a recursive method. As applications of these results, we derive the explicit formula to the eigentime identity for random walks on the hypercubes and show that it grows linearly with the network order. Moreover, we compute the number of spanning trees and the degree-Kirchhoff index of the hypercubes. Finally, we study the susceptible–infectious–susceptible (SIS) dynamics on the hypercubes and determine the epidemic threshold based on the spectral radius of the adjacency matrix. Throughout this paper, two numerical experiments are conducted based on the dynamics of complex networks, namely, random walks and epidemic spreading, and the results are consistent with our theoretical analysis.