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

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

2011-05-13

arxiv.org/abs/1105.2675v1

Mathematics

Combinatorics

28 pages, 3 figures

Scientific paper

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.

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