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 \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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2012
- DOI:
- 10.48550/arXiv.1206.5420
- arXiv:
- arXiv:1206.5420
- Bibcode:
- 2012arXiv1206.5420D
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Metric Geometry;
- 51M20;
- 52B15
- E-Print:
- Ars Mathematica Contemporanea (to appear, 29 pages)