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 G(G) of the symmetric group S p and then construct a vertex-transitive simple polytope of rank q, the graphicahedron, whose 1-skeleton (edge graph) is G(G). 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.

Symmetric graphicahedra

“…The mark on a triangle S of C is the image under π of the unique element ofà 2 that maps S to S , as illustrated in Figure 3. In particular, the mark on S is the identity permutation, denoted (1). Note that the chambers in C are not regular simplices when q > 3.…”

“…See Figure 3 for an illustration of the case q = 3, when P C3 is just the regular toroidal map {6, 3} (1,1) (see [1]). When q ≥ 4 the graphicahedron P Cq is not a regular polytope.…”

“…In [1,Section 4] it is shown that P G , with a (−1)-face appended, is an abstract q-polytope. Note that, if (K, α) and (L, β) are two faces of…”

“…We now briefly review the construction of the graphicahedron P G , an abstract polytope associated with a finite graph G (see [1] for details).…”

“…In the present paper, we continue the investigation of [1] of an interesting class of abstract polytopes, called graphicahedra, which are associated with graphs. Any finite connected graph G determines a G-graphicahedron.…”