Groups, Geometry, and Dynamics


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Volume 7, Issue 3, 2013, pp. 591–615
DOI: 10.4171/GGD/199

Published online: 2013-08-27

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

Ioannis Ivrissimtzis[1] and Norbert Peyerimhoff[2]

(1) University of Durham, UK
(2) University of Durham, UK

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.

Keywords: Vertex transitive graphs, Archimedean solids, Coxeter groups, random walks

Ivrissimtzis Ioannis, Peyerimhoff Norbert: Spectral representations of vertex transitive graphs, Archimedean solids and finite Coxeter groups. Groups Geom. Dyn. 7 (2013), 591-615. doi: 10.4171/GGD/199