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

Abstract

In this article, we study eigenvalue functions of varying transitionprobability matrices on finite, vertex transitive graphs. We provethat the eigenvalue function of an eigenvalue of fixed highermultiplicity has a critical point if and only if the correspondingspectral representation is equilateral. We also show how thegeometric realisation of a finite Coxeter group as a reflectiongroup can be used to obtain an explicit orthogonal system ofeigenfunctions. Combining both results, we describe the behaviour ofthe spectral representations of the second highest eigenvaluefunction under the change of the transition probabilities in thecase of Archimedean solids.