Spectral Properties of Extended Sierpiński Graphs and Their Applications

Abstract

The eigenvalues of a graph present a wide range of applications in structural and dynamical aspects of the graph. Determining and analyzing spectra of a graph has been an important and exciting research topic in recent years. In this paper, we study the spectra and their applications for extended Sierpinski graphs, which are closely related to WK-recursive networks that are widely used in the design and implementation of local area networks and parallel processing architectures. Moreover, a particular case of extended Sierpinski graphs is the dual of Apollonian network, which displays the prominent scale-free small-world characteristics as observed in various real networks. We derive recursive relations of the characteristic polynomials for extended Sierpinski graphs at two successive iterations, based on which we determine all the eigenvalues, their corresponding multiplicities and properties. We then use the obtained eigenvalues to evaluate the number of spanning trees, Kirchhoff index of extended Sierpinski graphs, as well as mean hitting time and cover time for random walks on the graphs.