Spectra of three-peg Hanoi towers graphs

TL;DR

This paper establishes a spectral decimation method to compute the Laplacian spectra of certain planar graphs related to Hanoi towers, connecting group theory and fractal analysis, and providing explicit eigenfunctions.

Contribution

It introduces a spectral decimation approach that simplifies spectrum calculation for graphs from self-similar groups and fractals, extending previous results.

Findings

Explicit spectrum of the graphs was computed

Eigenfunctions were fully described

Method bridges group theory and fractal analysis

Abstract

We consider the relationship between the Laplacians on two sequences of planar graphs, one from the theory of self-similar groups and one from analysis on fractals. By establishing a spectral decimation map between these sequences we give an elementary calculation of the spectrum of the former, which was first computed by Grigorchuk and \v{S}uni\'{c}. Our method also gives a full description of the eigenfunctions.