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Abstract:
In a natural way, the faces of ranks 1 and 2 in a 4-polytope P provide the vertices of a bipartite graph G. Recently,
Asia Weiss and I have examined this construction when P is a finite, abstract regular (or chiral) polytope of Schlafli type {3, q, 3}.
If, in this case, P is also self-dual, then G must be a 3-transitive (or 2-transitive) trivalent graph. With Egon Schulte and Tomaì Pisanski,
we have also proved that if P is not self-dual, then G is no more symmetric then it has right to be. Indeed, G is then a trivalent semisymmetric graph,
so that Aut( G ) is transitive on edges but not on vertices. (Such graphs are a little elusive.)
After covering some backgound ideas, I'll illustrate the theorems through some beautiful examples: for example, when P is
the 4-simplex (which of course can realized as a
regular convex polytope), the graph G is the Levi graph for the Desargues configuration.
And when P is the universal, locally toroidal abstract regular polytope
{ {3,6}_{(3,0)} , {6,3}_{(1,1)} } ,
we find that G is the Gray graph, whose 27 + 27 nodes make it the smallest semisymmetric trivalent graph.
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