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In this article, we study eigenvalue functions of varying transitionprobability matrices on finite, vertex transitive graphs. We prove that theeigenvalue function of an eigenvalue of fixed higher multiplicity has acritical point if and only if the corresponding spectral representation isequilateral. We also show how the geometric realisation of a finite Coxetergroup as a reflection group can be used to obtain an explicit orthogonal systemof eigenfunctions. Combining both results, we describe the behaviour of thespectral representations of the second highest eigenvalue function under thechange of the transition probabilities in the case of Archimedean solids.

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