Vibration Spectra of the $m$-Tree Fractal

TL;DR

This paper studies the vibrational properties of a new family of fractal trees with multiple branches, using spectral decimation to analyze their Laplacian spectra and how these spectra evolve with increasing branches.

Contribution

It introduces a novel class of post-critically finite fractal trees and develops a spectral analysis method for their Laplacians, extending understanding of fractal vibrational spectra.

Findings

Spectral decimation effectively describes the Laplacian spectrum.

Spectrum behavior analyzed as the number of branches increases.

New insights into vibrational properties of multi-branch fractal trees.

Abstract

We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases.