Another Proof of Oscar Rojo's Theorems

TL;DR

This paper provides an alternative proof of Oscar Rojo's theorems concerning the spectrum of the graph Laplacian on specific balanced trees, utilizing symmetry and eigenfunction analysis.

Contribution

It introduces a new proof method based on symmetry properties and eigenfunctions, offering a different perspective on Rojo's theorems.

Findings

Validates Rojo's theorems through symmetry-based proof

Highlights the role of eigenfunctions in spectral analysis

Provides insights into the structure of Laplacian spectra on balanced trees

Abstract

We present here another proof of Oscar Rojo's theorems about the spectrum of graph Laplacian on certain balanced trees, by taking advantage of the symmetry properties of the trees in question, and looking into the eigenfunctions of Laplacian.