Medial layer graphs of equivelar 4-polytopes

Semisymmetric graphs from polytopes

“…In Theorem 5 of [22] we find that G is 2-transitive when P is finite, chiral and self-dual of type {3, q, 3}; more specifically, G is then of type 2 + (respectively 2 − ) if and only if P is properly (respectively improperly) self-dual.…”

“…We refer to [22] or [25,Theorem 1] for the details of the construction. The directly regular case occurs if and only if Λ admits an involutory automorphism ρ such that (…”

“…Then one may construct a regular 4-polytope P = P(Γ ), of type {p 1 , p 2 , p 3 }, with Aut(P) = Γ . We refer to [22,Definition 2] or [23, Theorems 2E11 and 2E12] for details of the construction. Note also that P is self-dual if and only if Aut(P) admits an involutory group automorphism δ such that δρ j δ = ρ 3−j for j = 0, 1, 2, 3.…”

“…In [22], two of the authors established some general and unexpected connections between the two subjects, building upon a rich variety of examples appearing in the literature (see [4,7,10,11,28,29]). Here we develop these connections a little further, with specific focus on semisymmetric graphs.…”

Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators

“…A sequel to this paper will deal with (3) the idea of consistent cycles in graphs (see [1,2]) and its applications to maps, polytopes and maniplexes, and (4) a construction which expands the possibilities for semisymmetric graphs as medial layer graphs (see [3,4]). …”