Symmetric Graphicahedra

TL;DR

This paper investigates the symmetry properties of G-graphicahedra, a class of vertex-transitive polytopes derived from graphs, with detailed analysis on star and cycle graphs and their geometric relations.

Contribution

It provides a detailed analysis of the combinatorial and geometric symmetry properties of G-graphicahedra, especially for star and cycle graphs, linking them to Euclidean Coxeter groups and tessellations.

Findings

G-graphicahedra are vertex-transitive simple abstract polytopes.

Cycle graph G-graphicahedra relate to Euclidean Coxeter groups and torus tessellations.

The paper characterizes automorphism groups and symmetry properties of these polytopes.

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 \tilde{A}_{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.