Odgovori na vprašanja v zvezi z medialnimi slojnimi grafi samo-dualnih in kiralnih politopov

An abstract n-polytope P is a partially-ordered set which captures important properties of a geometric polytope, for any dimension n. For even n ≥ 2, the incidences between elements in the middle two layers of the Hasse diagram of P give rise to the medial layer graph of P, denoted by G = G(P). If n = 4, and P is both highly symmetric and self-dual of type {p, q, p}, then a Cayley graph C covering G can be constructed on a group of polarities of P. In this paper we address some open questions about the relationship between G and C that were raised in a 2008 paper by Monson and Weiss, and describe some interesting examples of these graphs. In particular, we give the first known examples of improperly self-dual chiral polytopes of type {3, q, 3}, which are also among the very few known examples of highly symmetric self-dual finite polytopes that do not admit a polarity. Also we show that if p = 3 then C cannot have a higher degree of s-arc-transitivity than G, and we present a family of regular 4-polytopes of type {6, q, 6} for which the vertex-stabilisers in the automorphism group of C are larger than those for G.