  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.
