Extended Vicsek fractals: Laplacian spectra and their applications

TL;DR

This paper analyzes the Laplacian spectra of extended Vicsek fractals with linear spacers, providing analytic expressions for spectral sums that relate to large-scale properties and dynamics of these complex structures.

Contribution

It introduces recurrence relations for Laplacian spectra of EVF and derives an analytic expression for the sum of inverse nonvanishing eigenvalues, linking structure to dynamics.

Findings

Analytic expression for spectral sum of EVF

Linear spacers induce local heterogeneities

Spectral properties relate to polymer dynamics

Abstract

Extended Vicsek fractals (EVF) are the structures constructed by introducing linear spacers into traditional Vicsek fractals. Here we study the Laplacian spectra of the EVF. In particularly, the recurrence relations for the Laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing Laplacian eigenvalues. This quantity characterizes the large-scale properties, such as the gyration radius of the polymeric structures, or the global mean-first passage time for the random walk processes. Introduction of the linear spacers leads to local heterogeneities, which reveal themselves, for example, in the dynamics of EVF under external forces.