Dual complementary polynomials of graphs and combinatorial-geometric interpretation on the values of Tutte polynomial at positive integers

• Authors: Beifang Chen
• Venue: European Journal of Combinatorics
• Year: 2014

We introduce modular (integral) complementary polynomial @k (@k"Z) of two variables on a graph G by counting the number of modular (integral) complementary tension-flows. We further introduce cut-Eulerian equivalence relation on orientations and geometric structures: complementary open lattice polyhedron @D"c"t"f, 0-1 polytope @D"c"t"f^+, and lattice polytopes @D"c"t"f^@r with respect to orientations @r. The polynomial @k (@k"Z) is a common generalization of the modular (integral) tension polynomial @t (@t"Z) and the modular (integral) flow polynomial @f (@f"Z) of one variable, and can be decomposed into a sum of product Ehrhart polynomials of complementary open 0-1 polytopes. There are dual complementary polynomials @k@? and @k@?"Z, dual to @k and @k"Z respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes @D@?"c"t"f^+. It turns out remarkably that @k@? is Whitney's rank generating polynomial R"G, which gives rise to a nontrivial combinatorial-geometric interpretation on the values of the Tutte polynomial T"G at all positive integers. In particular, some special values of @k"Z and @k@?"Z (@k and @k@?) count the number of certain special kinds (of equivalence classes) of orientations, including the recovery of a few well-known values of T"G.