JournalsggdVol. 7, No. 3pp. 591–615

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

  • Ioannis Ivrissimtzis

    University of Durham, UK
  • Norbert Peyerimhoff

    University of Durham, UK
Spectral   representations of vertex transitive graphs, Archimedean solids and   finite Coxeter groups cover
Download PDF

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.

Cite this article

Ioannis Ivrissimtzis, Norbert Peyerimhoff, Spectral representations of vertex transitive graphs, Archimedean solids and finite Coxeter groups. Groups Geom. Dyn. 7 (2013), no. 3, pp. 591–615

DOI 10.4171/GGD/199