Spectral analysis for weighted extended Vicsek polygons

Abstract

Because of the application of fractal networks and their spectral properties in various fields of science and engineering, they have become a hot topic in network science. Moreover, deterministic weighted graphs are widely used to model complex real-world systems. This paper studys weighted extended Vicsek polygons $W(G_{m,t})$, which are based on the Vicsek fractal model and the extended fractal cactus model. The structure of these polygons is controlled by the positive integer coefficient $m$ and the number of iterations $t$. From the construction of the graph, we derive recursive relations of all eigenvalues and their multiplicities of normalized Laplacian matrices from the two successive generations of the weighted extended Vicsek polygons. Then, we use the spectra of the normalized Laplacian matrices to study Kemeny's constant, the multiplicative Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for the weighted extended Vicsek polygons. The above results help to analyze the topology and dynamic properties of the network model, so it has potential application prospects.