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

Abstract

We introduce modular (integral) complementary polynomial κ (κZ) 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 Δctf, 0–1 polytope Δctf+, and lattice polytopes Δctfρ with respect to orientations ρ. The polynomial κ (κZ) is a common generalization of the modular (integral) tension polynomial τ (τZ) and the modular (integral) flow polynomial φ (φZ) 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 κ̄ and κ̄Z, dual to κ and κZ respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes Δ̄ctf+. It turns out remarkably that κ̄ is Whitney’s rank generating polynomial RG, which gives rise to a nontrivial combinatorial–geometric interpretation on the values of the Tutte polynomial TG at all positive integers. In particular, some special values of κZ and κ̄Z (κ and κ̄) count the number of certain special kinds (of equivalence classes) of orientations, including the recovery of a few well-known values of TG.