arXiv 1105.26752011-05math.CO

图的对偶互补多项式及Tutte多项式在正整数处值的组合解释

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

Beifang Chen

备注：: 28 pages, 3 figures

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摘要：

通过计算图$G$的有向模（整）补张力流（CTF）的个数，引入了图$G$的二元模（整）补多项式$κ(G;x,y)$（$κ_{\mathbbm z}(G;x,y)$）.我们研究这些多项式通过进一步引入一个切欧拉等价关系的方向和几何结构，如互补的开格多面体$Δ_\textsc{ctf}(G,ε)$，互补的开0-1多面体$Δ^+_\textsc{ctf}(G,ε)$，和互补的开格多面体$Δ^ρ_\textsc{ctf}(G,ε)$相对于方向$ρ$。多项式$κ(G;x,y)$（$κ_{\mathbbm z}(G;x,y)$）是模（积分）张力多项式$τ(G,x)$（$τ_\mathbbm{z}(G,x)$）和模（积分）流多项式$ϕ(G,y)$（$ϕ_\mathbbm{z}(G,y)$）的通用推广，并且可以分解为互补开0-1多面体$Δ^+_\textsc{ctf}(G,ρ)$的乘积Ehrhart多项式之和。存在对偶互补多项式$\barκ(G;x,y)$和$\barκ_{\mathbbm z}(G;x,y)$，分别与$κ$和$κ_{\mathbbm z}$对偶，在这个意义上，对Ehrhart多项式的格点计数取在扩张的闭多面体$\barΔ^+_\textsc{ctf}(G,ρ)$的拓扑和内。事实证明，多项式$\barκ(G;x,y)$是惠特尼的秩生成多项式$R_G(x,y)$，这引起了对Tutte多项式$T_G(x,y)$在正整数的值的组合解释。特别地，$κ_\mathbbm{z}$和$\barκ_\mathbbm{z}$的一些特殊值（$κ$和$\barκ$）计算某些特殊种类（等价类）的方向的数量。

英文摘要：

We introduce a modular (integral) complementary polynomial $κ(G;x,y)$ ($κ_{\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $ε$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $Δ_\textsc{ctf}(G,ε)$, the complementary open 0-1 polytope $Δ^+_\textsc{ctf}(G,ε)$, and the complementary open lattice polytopes $Δ^ρ_\textsc{ctf}(G,ε)$ with respect to orientations $ρ$. The polynomial $κ(G;x,y)$ ($κ_{\mathbbm z}(G;x,y)$) is a common generalization of the modular (integral) tension polynomial $τ(G,x)$ ($τ_\mathbbm{z}(G,x)$) and the modular (integral) flow polynomial $ϕ(G,y)$ ($ϕ_\mathbbm{z}(G,y)$), and can be decomposed into a sum of product Ehrhart polynomials of complementary open 0-1 polytopes $Δ^+_\textsc{ctf}(G,ρ)$. There are dual complementary polynomials $\barκ(G;x,y)$ and $\barκ_{\mathbbm z}(G;x,y)$, dual to $κ$ and $κ_{\mathbbm z}$ respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes $\barΔ^+_\textsc{ctf}(G,ρ)$. It turns out that the polynomial $\barκ(G;x,y)$ 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 $κ_\mathbbm{z}$ and $\barκ_\mathbbm{z}$ ($κ$ and $\barκ$) count the number of certain special kinds (of equivalence classes) of orientations.

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