2020/09/04 by Winter, Martin
#05C50 #05C62 #51M20 #52B05 #52B11 #52B12 #52B15 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2009.02179
Starting from a finite simple graph G, for each eigenvalue θ of its adjacency matrix one can construct a convex polytope PG(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, θ2-spectral). We apply this criterion to the edge-transitive polytopes. We show that every edge-transitive polytope is θ2-spectral, is uniquely determined by this graph, and realizes all its symmetries. We give a complete classification of distance-transitive polytopes.