Symmetric graphicahedra

Abstract

Given a connected graph G with p vertices and q edges, the G-graphicahedron is a vertex-transitive simple abstract polytope of rank q whose edge-graph is isomorphic to a Cayley graph of the symmetric group S_p associated with G. The paper explores combinatorial symmetry properties of G-graphicahedra, focussing in particular on transitivity properties of their automorphism groups. We present a detailed analysis of the graphicahedra for the q-star graphs K_1, q and the q-cycles C_q. The C_q-graphicahedron is intimately related to the geometry of the infinite Euclidean Coxeter group Ã_{q − 1} and can be viewed as an edge-transitive tessellation of the (q − 1)-torus by (q − 1)-dimensional permutahedra, obtained as a quotient, modulo the root lattice A_{q − 1}, of the Voronoi tiling for the dual root lattice A_{q − 1} ^* in Euclidean (q − 1)-space.