The Graphicahedron

TL;DR

This paper introduces the graphicahedron, a new class of abstract polytopes derived from Cayley graphs of symmetric groups, generalizing permutahedra and exploring their symmetry and structure.

Contribution

It presents a novel construction of abstract polytopes from Cayley graphs, extending the permutahedron concept to arbitrary connected graphs.

Findings

The graphicahedron is a vertex-transitive simple polytope.

Its structure is explicitly determined for small graphs.

The paper discusses symmetry properties of the graphicahedron.

Abstract

The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph of the symmetric group S_p and then construct a vertex-transitive simple polytope of rank q, called the graphicahedron, whose 1-skeleton (edge graph) is the Cayley graph. The graphicahedron of a graph G is a generalization of the well-known permutahedron; the latter is obtained when the graph is a path. We also discuss symmetry properties of the graphicahedron and determine its structure when G is small.