Eigenpolytopes, Spectral Polytopes and Edge-Transitivity

Starting from a finite simple graph $G$, for each eigenvalue $θ$ of its adjacency matrix one can construct a convex polytope $P_G(θ)$, the so called $θ$-eigenpolytop of $G$. For some polytopes this technique can be used to reconstruct the polytopes from its edge-graph. Such polytopes (we shall call them spectral) are still badly understood. We give an overview of the literature for eigenpolytopes and spectral polytopes. We introduce a geometric condition by which to prove that a given polytope is spectral (more exactly, $θ_2$-spectral). We apply this criterion to the edge-transitive polytopes. We show that every edge-transitive polytope is $θ_2$-spectral, is uniquely determined by this graph, and realizes all its symmetries. We give a complete classification of distance-transitive polytopes.

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