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

Ivrissimtzis, Ioannis; Peyerimhoff, Norbert

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.

Citation

Ivrissimtzis, I., & Peyerimhoff, N. (2013). Spectral representations of vertex transitive graphs, Archimedean solids and finite Coxeter groups. Groups, Geometry, and Dynamics, 7(3), 591-615. https://doi.org/10.4171/ggd/199

• Journal Article Type: Article
• Acceptance Date: Mar 19, 2012
• Online Publication Date: Aug 27, 2013
• Publication Date: Aug 27, 2013
• Deposit Date: Jun 17, 2019
• Publicly Available Date: Jun 17, 2019
• Journal: Groups, Geometry, and Dynamics
• Print ISSN: 1661-7207
• Publisher: EMS Press
• Peer Reviewed: Peer Reviewed
• Volume: 7
• Issue: 3
• Pages: 591-615
• DOI: https://doi.org/10.4171/ggd/199