完全图的Toroidal链的正合色多项式
Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs
摘要：
本文给出了零温Potts反铁磁体的配分函数（等价于色多项式）的精确计算，该配分函数是由在纵向上具有周期或扭曲周期边界条件的重复完全子图$K_b$和$b=5,6$组成的任意长的图.在$L_x \to \infty$极限中，确定色零点${\cal B}$的连续累积集。本文给出了任意$b$的一些结果，包括以$b-1$为度系数的特征值的极值和几类特征值的显式。证明了${\cal B}$与实轴相交的极大点$q_c$满足$2 \le b$的$q_c \le b$不等式，${\cal B}$与$q$实轴相交的$q$的极小值为$q=0$，并对Klein瓶带色多项式的结构作了一个猜想.
英文摘要：
We present exact calculations of the partition function of the zero-temperature Potts antiferromagnet (equivalently, the chromatic polynomial) for graphs of arbitrarily great length composed of repeated complete subgraphs $K_b$ with $b=5,6$ which have periodic or twisted periodic boundary condition in the longitudinal direction. In the $L_x \to \infty$ limit, the continuous accumulation set of the chromatic zeros ${\cal B}$ is determined. We give some results for arbitrary $b$ including the extrema of the eigenvalues with coefficients of degree $b-1$ and the explicit forms of some classes of eigenvalues. We prove that the maximal point where ${\cal B}$ crosses the real axis, $q_c$, satisfies the inequality $q_c \le b$ for $2 \le b$, the minimum value of $q$ at which ${\cal B}$ crosses the real $q$ axis is $q=0$, and we make a conjecture concerning the structure of the chromatic polynomial for Klein bottle strips.
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