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

TL;DR

This paper introduces dual complementary polynomials of graphs, linking them to the Tutte polynomial and providing combinatorial interpretations of their values at positive integers through geometric and lattice point counting methods.

Contribution

It defines new modular and integral complementary polynomials, relates them to existing polynomials, and connects their duals to the Tutte polynomial with geometric and combinatorial insights.

Findings

The polynomial arppa(G;x,y) equals Whitney's rank generating polynomial R_G(x,y).

Special values of the polynomials count certain classes of orientations.

Decomposition of ppa(G;x,y) into Ehrhart polynomials of polytopes.

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.