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.