Eigenvalues of Laplacians on Higher Dimensional Vicsek Set Graphs

TL;DR

This paper investigates the eigenvalues of Laplacians on higher-dimensional Vicsek set graphs, extending previous two-dimensional results and analyzing lattice isomorphisms.

Contribution

It introduces a spectral decimation function for higher-dimensional Vicsek set graphs and provides criteria for lattice isomorphisms, advancing understanding of their spectral properties.

Findings

Derived a general spectral decimation function for higher-dimensional Vicsek graphs

Extended eigenvalue analysis from 2D to higher dimensions

Established criteria for Vicsek set lattice isomorphisms

Abstract

We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general spectral decimation function. This is an extension of earlier results on two dimensional Vicsek sets. Second, we study the Vicsek set lattices, which are natural analogues to the Sierpinski lattices. We have a criterion when two different Vicsek set lattices are isomorphic.