Geodesic
Spectral representations of vertex transitive graphs, Archimedean solids and finite Coxeter groups
Ioannis Ivrissimtzis1 ; Norbert Peyerimhoff1 orcid
1University of Durham, UK
Groups, geometry, and dynamics, Tome 7 (2013) no. 3, pp. 591-615

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DOI

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.
DOI : 10.4171/ggd/199
Classification : 20-XX, 05-XX, 52-XX
Mots-clés : Vertex transitive graphs, Archimedean solids, Coxeter groups, random walks

Ioannis Ivrissimtzis  1   ; Norbert Peyerimhoff  1

1 University of Durham, UK 
Ioannis Ivrissimtzis; Norbert Peyerimhoff. Spectral   representations of vertex transitive graphs, Archimedean solids and   finite Coxeter groups. Groups, geometry, and dynamics, Tome 7 (2013) no. 3, pp. 591-615. doi: 10.4171/ggd/199
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     author = {Ioannis Ivrissimtzis and Norbert Peyerimhoff},
     title = {Spectral   representations of vertex transitive graphs, {Archimedean} solids and   finite {Coxeter} groups},
     journal = {Groups, geometry, and dynamics},
     pages = {591--615},
     year = {2013},
     volume = {7},
     number = {3},
     doi = {10.4171/ggd/199},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/199/}
}
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