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

Abstract

We introduce a modular (integral) complementary polynomial $\kappa(G;x,y)$ ($\kappa_{\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $\epsilon$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $\Delta_\textsc{ctf}(G,\epsilon)$, the complementary open 0-1 polytope $\Delta^+_\textsc{ctf}(G,\epsilon)$, and the complementary open lattice polytopes $\Delta^\rho_\textsc{ctf}(G,\epsilon)$ with respect to orientations $\rho$. The polynomial $\kappa(G;x,y)$ ($\kappa_{\mathbbm z}(G;x,y)$) is a common generalization of the modular (integral) tension polynomial $\tau(G,x)$ ($\tau_\mathbbm{z}(G,x)$) and the modular (integral) flow polynomial $\phi(G,y)$ ($\phi_\mathbbm{z}(G,y)$), and can be decomposed into a sum of product Ehrhart polynomials of complementary open 0-1 polytopes $\Delta^+_\textsc{ctf}(G,\rho)$. There are dual complementary polynomials $\bar\kappa(G;x,y)$ and $\bar\kappa_{\mathbbm z}(G;x,y)$, dual to $\kappa$ and $\kappa_{\mathbbm z}$ respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes $\bar\Delta^+_\textsc{ctf}(G,\rho)$. It turns out that the polynomial $\bar\kappa(G;x,y)$ is Whitney's rank generating polynomial $R_G(x,y)$, which gives rise to a combinatorial interpretation on the values of the Tutte polynomial $T_G(x,y)$ at positive integers. In particular, some special values of $\kappa_\mathbbm{z}$ and $\bar\kappa_\mathbbm{z}$ ($\kappa$ and $\bar\kappa$) count the number of certain special kinds (of equivalence classes) of orientations.