Spectral representations of vertex transitive graphs, Archimedean solids and finite Coxeter groups

TL;DR

This paper investigates the spectral properties of vertex transitive graphs, especially Archimedean solids, revealing conditions for critical points of eigenvalue functions and constructing explicit eigenfunctions using Coxeter groups.

Contribution

It establishes a link between spectral representations and geometric symmetries, providing explicit eigenfunctions for finite Coxeter groups and analyzing eigenvalue behavior in Archimedean solids.

Findings

Eigenvalue functions have critical points iff spectral representations are equilateral.

Explicit orthogonal eigenfunctions are constructed via Coxeter group realizations.

Behavior of second highest eigenvalue under transition probability changes is characterized.

Abstract

In this article, we study eigenvalue functions of varying transition probability matrices on finite, vertex transitive graphs. We prove that the eigenvalue function of an eigenvalue of fixed higher multiplicity has a critical point if and only if the corresponding spectral representation is equilateral. We also show how the geometric realisation of a finite Coxeter group as a reflection group can be used to obtain an explicit orthogonal system of eigenfunctions. Combining both results, we describe the behaviour of the spectral representations of the second highest eigenvalue function under the change of the transition probabilities in the case of Archimedean solids.