平面三角剖分图的色多项式的结构及其与色零点和渐近极限的关系
The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities
摘要：
本文研究了平面三角剖分图$G_{pt,\vec m}$的单参数和多参数族的色多项式$P(G_{pt,\vec m},q)$的结构和性质，其中${\vec m} = (m_1,...,m_p)$是整数参数向量.我们使用这些来研究$|P(G_{pt,\vec m},τ+1)|$与Tutte上限$(τ-1)^{n-5}$的比率，其中$τ=(1+\sqrt{5} \ )/2$和$n$是$G_{pt,\vec m}$中的顶点数。特别是，我们计算的极限值，这个比例为PDMATHAMT各种家庭的平面三角剖分。我们也用我们的计算来研究这些色多项式的零点。我们研究了一个大类的家庭$G_{pt,\vec m}$与$p=1$和$p=2$，并表明，这些有一个结构的形式$P(G_{pt,m},q) = c_{_{G_{pt}},1}λ_1^m + c_{_{G_{pt}},2}λ_2^m + c_{_{G_{pt}},3}λ_3^m$的$p=1$，其中$λ_1=q-2$，$λ_2=q-3$，和$λ_3=-1$，和$P(G_{pt,\vec m},q) = \sum_{i_1=1}^3 \sum_{i_2=1}^3 c_{_{G_{pt}},i_1 i_2} λ_{i_1}^{m_1}λ_{i_2}^{m_2}$的$p=2$。我们推导出的系数$c_{_{G_{pt}},\vec i}$的属性，并表明$P(G_{pt,\vec m},q)$有一个真正的色零点，接近$(1/2)(3+\sqrt{5} \ )$作为一个或多个的$m_i \to \infty$。给出了对$p \ge 3$的推广。此外，我们提出了一个单参数家庭的平面三角剖分与实零点，从下面的$m \to \infty$接近3。的Potts反铁磁体的基态熵的影响进行了讨论。
英文摘要：
We present an analysis of the structure and properties of chromatic polynomials $P(G_{pt,\vec m},q)$ of one-parameter and multi-parameter families of planar triangulation graphs $G_{pt,\vec m}$, where ${\vec m} = (m_1,...,m_p)$ is a vector of integer parameters. We use these to study the ratio of $|P(G_{pt,\vec m},τ+1)|$ to the Tutte upper bound $(τ-1)^{n-5}$, where $τ=(1+\sqrt{5} \ )/2$ and $n$ is the number of vertices in $G_{pt,\vec m}$. In particular, we calculate limiting values of this ratio as $n \to \infty$ for various families of planar triangulations. We also use our calculations to study zeros of these chromatic polynomials. We study a large class of families $G_{pt,\vec m}$ with $p=1$ and $p=2$ and show that these have a structure of the form $P(G_{pt,m},q) = c_{_{G_{pt}},1}λ_1^m + c_{_{G_{pt}},2}λ_2^m + c_{_{G_{pt}},3}λ_3^m$ for $p=1$, where $λ_1=q-2$, $λ_2=q-3$, and $λ_3=-1$, and $P(G_{pt,\vec m},q) = \sum_{i_1=1}^3 \sum_{i_2=1}^3 c_{_{G_{pt}},i_1 i_2} λ_{i_1}^{m_1}λ_{i_2}^{m_2}$ for $p=2$. We derive properties of the coefficients $c_{_{G_{pt}},\vec i}$ and show that $P(G_{pt,\vec m},q)$ has a real chromatic zero that approaches $(1/2)(3+\sqrt{5} \ )$ as one or more of the $m_i \to \infty$. The generalization to $p \ge 3$ is given. Further, we present a one-parameter family of planar triangulations with real zeros that approach 3 from below as $m \to \infty$. Implications for the ground-state entropy of the Potts antiferromagnet are discussed.
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