Twisted duality and isomorphism of medial graphs
|University of South Alabama
|Mathematics & Computer Building (MC) 5158
This talk revolves around two fundamental constructions in graph theory: duals and medial graphs. There are a host of well-known relations between the duals and the medial graphs of plane graphs. By considering these relations we will be led to the principle that duality and equality of plane graphs are equivalent concepts. We will then examine what happens when we change our notion of equivalence of embedded graphs. In particular, we will see how isomorphism of abstract graphs corresponds to an extension of duality called twisted duality, and how twisted duality extends fundamental relations between duals and medial graphs from plane graphs to graphs embedded in other surfaces. We will then go on to see how this group action leads to a deeper understanding of the properties of, and relationships among, various graph polynomials including the chromatic polynomial, the Penrose polynomial, and topological Tutte polynomials.