Symmetric Graphs from Polytopes of High Rank

Abstract

Given a polytope \({{\mathcal{P}}}\) of rank 2n, the faces of middle ranks n − 1 and n constitute the vertices of a bipartite graph, the medial layer graph \({{M(\mathcal{P})}}\) of \({{\mathcal{P}}}\). The group \({{D(\mathcal{P})}}\) of automorphisms and dualities of \({{\mathcal{P}}}\) has a natural action on this graph. We prove algebraic and combinatorial conditions on \({{\mathcal{P}}}\) that ensure this action is transitive on k-arcs in \({{M(\mathcal{P})}}\) for some small k (in particular focussing on k = 3), and provide examples of families of polytopes that satisfy these conditions. We also examine how \({{D(\mathcal{P})}}\) acts on the k-stars based at vertices of \({{M(\mathcal{P})},}\) and describe self-dual regular polytopes (in particular those of rank 6) for which this action is transitive on the k-stars for small k.