七个苯环生成的六角系统的双强迫多项式

Double Forcing Polynomials of a Hexagonal System Generated by Seven Benzene Rings

摘要: 匹配是一个边的集合，其中任意两条边都没有公共顶点。对于图G的一个匹配M，如果M中的边能够将G的所有顶点两两配对，则称该匹配为完美匹配。七个苯环生成的六角系统中具有完美匹配的六角系统个数为190个。本文计算出了这190个七个苯环生成的六角系统的双强迫多项式。同时将双强迫多项式、强迫多项式、反强迫多项式、完美匹配个数、自由度与反自由度对于图的区分情况进行了统计与比较。

Abstract: A matching is a set of edges, where any two edges have no common vertices. For a match M in graph G, if the edges in M can pair all the vertices of G in pairs, the match is said to be a perfect match. The number of hexagonal systems with perfect matchings among the hexagonal systems generated by seven benzene rings is 190. This paper calculates the di-forcing polynomials of the hexagonal system generated by these 190 seven benzene rings. At the same time, the discrimination of di-forcing polynomials, forced polynomials, anti-forced polynomials, number of perfect matches, degrees of freedom and anti-degrees of freedom for graphs is statistically compared.

1. 引言

完美匹配在有机化学中有着重要应用，化学分子的凯库勒结构就是图的完美匹配。凯库勒结构与分子的稳定性密切相关。Lovász等[1]系统的总结了匹配理论的研究背景与早期发展成果。

凯库勒结构的内自由度概念是由Randic和Klein [2] [3]在研究有机化学分子时首次提出，随后Harary [4]在研究六角系统时正式将其命名为强迫数。为了更好地研究图G的强迫数分布，P. Adams，M. Mahdian和E. Mahmoodian [5]定义了图G的强迫谱$Spe{c}_f\left(G\right)$ ，它是G中所有完美匹配强迫数所构成的集合。为了更加细致的刻画所有完美匹配的强迫数，张和平和赵爽[6]等提出了图G的强迫多项式。

Vukicevic, D.和Trinajstic, N. [7]首次提出反强迫数的概念。雷洪川，张和平等[8] [9]引入了图的反强迫谱定义，即图G中所有完美匹配的反强迫数组成的集合，记为$Spe{c}_{af}\left(G\right)$ 。Hwang [10]等提出了图的反强迫多项式。

刘雨童，马聪聪，姚海元[11] [12]在强迫多项式和反强迫多项式的基础上首次提出双强迫多项式，用以更好全面的刻画完美匹配的一些性质。双强迫多项式通过赋值可以得到强迫多项式、反强迫多项式，以及求出完美匹配的个数，展现了双强迫多项式较高的应用价值。刘雨童[13]首先对双强迫多项式进行了应用，计算出了所有C₆₀富勒烯同分异构体的双强迫多项式。王彦通[14]对循环梯状图的双强迫多项式进行了研究。目前关于图的强迫多项式以及反强迫多项式研究取得了一些进展，但是对于双强迫多项式的研究成果较少。

P. Hansen和M. Zheng [15]以及张福基，李学良[16]都曾通过不同的方法独立刻画出了具有强迫边的六角系统。刘乙瑾[17]等枚举出六个苯环生成的六角系统，并算出了其强迫多项式与自由度。俞德龙[18]等计算出苯环数目不超过六的六角系统的双强迫多项式。邓凯[19]等枚举出190个具有完美匹配的七个苯环生成的六角系统，并计算出了其强迫多项式与反强迫多项式。本文在此基础上计算出了190个双强迫多项式，并比较了双强迫多项式、强迫多项式、反强迫多项式等对六角系统的区分度。

2. 预备知识

有向简单图G为一个有序二元组$\left(V\left(G\right),E\left(G\right)\right)$ ，将$V\left(G\right)$ 与$E\left(G\right)$ 分别表示图G的顶点构成的集合与图G的边构成的集合。M称为图G的匹配，如果M的元素均为G的边，且M中任意两个边在G中均不相邻(没有公共端点)。与匹配M中的边关联的顶点称作被M覆盖。匹配M覆盖图G的所有顶点时该匹配称为完美匹配。若M是图G的一个完美匹配，且M的子集S不包含在G的其他完美匹配中，则称S为M的一个强迫集。M的一个包含边数最小的强迫集称作最小强迫集。最小强迫集中元素的个数称作M的强迫数，记为$f\left(G,M\right)$ 。图G中所有完美匹配的强迫数的最大值，称作图G的最大强迫数，记为$F\left(G\right)$ ；所有完美匹配的强迫数的最小值，称作图G的最小强迫数，记为$f\left(G\right)$ 。图G完美匹配的强迫数之和称作图G的自由度，也叫内自由度，记为$IDF\left(G\right)$ 。图的一个完美匹配对应一个幂为该匹配强迫数，同时系数为1的单项式，将每一个完美匹配对应的单项式相加即为强迫多项式。强迫多项式记为$F\left(G,x\right)$ 。

设M为图G的一个完美匹配，令$S_a\subseteq E\left(G\right)\backslash M$ ，如果M是$G-S_a$ 中唯一的完美匹配，则称$S_a$ 为M的反强迫集，M的最小的反强迫集所包含的元素个数称作最小反强迫数，记为$af\left(G,M\right)$ 。图G中所有完美匹配的反强迫数的最大值，称作图G的最大反强迫数，记为$AF\left(G\right)$ ；所有完美匹配的反强迫数的最小值，称作图G的最小反强迫数，记为$af\left(G\right)$ 。图G完美匹配的反强迫数之和称作图G的反自由度，记为$ADF\left(G\right)$ 。图的一个完美匹配对应一个幂为该匹配反强迫数，同时系数为1的单项式，将每一个完美匹配对应的单项式相加即为反强迫多项式。反强迫多项式记为$Af\left(G,x\right)$ 。

定义 1 图G的双强迫多项式：

$Faf\left(G;x,y\right)={\displaystyle \sum _{i,j}\upsilon \left(G;i,j\right)x^i{y}^j}$

其中$\upsilon \left(G;i,j\right)$ 表示强迫数为i，反强迫数为j的完美匹配的个数。

推论1 [11] 在图G的双强迫多项式中，当多项式中的y全部赋值为1时，得到图G的强迫多项式；当多项式中的x全部赋值为1 (同时将y变为x)，得到图G的反强迫多项式。即：

$Faf\left(G;x,1\right)=F\left(G,x\right)={\displaystyle \sum _{i=f\left(G\right)}^{F\left(G\right)}\omega \left(G,i\right)x^i}$

$Faf\left(G;1,x\right)=Af\left(G,x\right)={\displaystyle \sum _{i=af\left(G\right)}^{AF\left(G\right)}\upsilon \left(G,i\right)x^i}$

$\omega \left(G,i\right)$ 表示图G强迫数为i的完美匹配个数；$\upsilon \left(G,i\right)$ 表示图G反强迫数为i的完美匹配个数。

推论2 [11] 在图G的双强迫多项式中，令x与y全部赋值为1，最终计算出的数值为图的完美匹配的个数。即：

$Faf\left(G;1,1\right)=\Phi \left(G\right)$

推论3 [11] 图G的强迫多项式求一阶导数，随后将x全部赋值为1，即可计算得到图G的自由度。由以上关系可引出图的双强迫多项式与自由度的之间的关系：

$IDF\left(G\right)={\frac{\partial }{\partial x}Faf\left(G;x,y\right)|}_{\left(x,y\right)\to \left(1,1\right)}$

推论4 [11] 图G的反强迫多项式求一阶导数，随后将y全部赋值为1，即可计算得到图G的反自由度。由以上关系可引出图的双强迫多项式与反自由度的之间的关系：

$ADF\left(G\right)={\frac{\partial }{\partial y}Faf\left(G;x,y\right)|}_{\left(x,y\right)\to \left(1,1\right)}$

图的双强迫多项式的求法有两种，即穷举法和整数线性规划法。穷举法需要找到图的所有完美匹配，再从每一个完美匹配中寻找强迫集和反强迫集，从而得到强迫数和反强迫数，最终写出双强迫多项式。穷举法过程复杂较为耗时。下面重点介绍整数线性规划规划法，利用线性规划法求解双强迫多项式。

首先，先生成图G的所有M-交错圈，即生成所有完美匹配中匹配边和非匹配边交替出现的圈。因为每一个连通分支必是M-交错圈，从而得到了所有的连通分支。令$N=\left(n_{ij}\right)$ 是一个$l\times n$ 的矩阵，其中l表示的是M-交错圈的数目，N的每个行向量$\left\{n_{i1},n_{i2},\cdots ,n_{in}\right\}$ 对应于一个M-交错圈$C_i$ 的0-1关联向量，$n_{ij}=1$ 当且仅当$e_j\in C_i\cap M$ ，$i=1,2,\cdots ,l$ ，$j=1,2,\cdots ,n$ 。称N为完美匹配M的强迫系数矩阵。令$b={\left(1,1,\cdots ,1\right)}^{\text{T}}$ ，$c={\left(1,1,\cdots ,1\right)}^{\text{T}}$ ，其b的维数为l，c的维数为n。因此有整数线性规划(ILP)：

$\begin{array}{l}\left(\text{ILP}\right):\min c^{\text{T}}x,\\ \text{s}\text{.t}.\{\begin{array}{l}Nx\ge b,\\ x\in {\left\{0,1\right\}}^n.\end{array}\end{array}$

(ILP)的最优值即为M的强迫数，最优解x即最小强迫集S的关联向量。恰恰相反，如果在(ILP)上非匹配边位置写1，其他位置写0。即可得到完美匹配的反强迫数。拥有了所有完美匹配及其强迫数与反强迫数，就可写出对应的双强迫多项式。

3. 七个苯环生成的六角系统

六角系统是一个2-连通平面二部图，其每个内面是一个单位正六边形，在化学领域研究中有着重要应用。在研究苯类碳氢化合物时，将氢原子忽略不计，苯类碳氢化合物的原子骨架图即对应一个六角系统。其中碳原子即为图中的点，碳原子与碳原子之间的键即为图的边，碳碳双键即为完美匹配的匹配边。化学中已经证明稳定的苯类碳氢化合物一定有凯库勒结构[20]。确定一个六角系统苯环的个数，不同构的六角系统完美匹配的个数就等于该数目的苯环生成的碳氢化合物同分异构体的个数。便可将化学问题转化为数学问题。六角系统可以划分为cata-型和peri-型两种，二者区别在于六角系统中是否存在三个六边形共用一个顶点。若图中没有三个六边形共用一个顶点，则六角系统为cata-型六角系统。cata-型六角系统一定有完美匹配。若图中存在一个或一个以上三个六边形共用一个顶点情况，则六角系统为peri-型六角系统。

刘乙瑾[17]等枚举出了由六个苯环生成的六角系统，并计算出六个苯环生成的六角系统的强迫多项式与自由度。俞德龙[18]等计算出了苯环数目不超过六的六角系统的双强迫多项式，并对双强迫多项式、强迫多项式和反强迫多项式的区分度进行了比较。

目前已知七个苯环生成的六角系统个数为331个，其中拥有完美匹配的六角系统的个数为190个[21]，cata-型六角系统有118个[22]，peri-型六角系统有72个。邓凯[19]等计算出了七个苯环生成的六角系统的强迫多项式和反强迫多项式，在此基础上，对七个苯环组成的六角系统双强迫多项式进行研究。

对于没有完美匹配的六角系统我们不再进行研究。文献[19]的附录给出了由七个苯环构成的所有190个有完美匹配的六角系统，同时该附录对190个有完美匹配的七个苯环的六角系统从H₁-H₁₉₀进行了编号。为了方便起见将计算结果按原顺序排列。

4. 主要结果

利用文献[19]的枚举结果，用整数线性规划法计算了由七个苯环生成的190个有完美匹配的六角系统的双强迫多项式，由此可得到完美匹配数，自由度与反自由度。并比较了由七个苯环生成的六角系统的双强迫多项式、强迫多项式、反强迫多项式、完美匹配个数、自由度与反自由度对七个苯环生成的六角系统的区分度。

下面首先来考察双强迫多项式，强迫多项式以及反强迫多项式对六角系统的区分情况。根据表1、表2及七个苯环生成的六角系统强迫多项式与反强迫多项式[19]构造出了其双强迫多项式，强迫多项式以及反强迫多项式关系图(图1)。其中红色实线表示两个六角系统双强迫多项式相同，绿色实线表示两个六角系统强迫多项式相同，蓝色实线表示两个六角系统反强迫多项式相同。

根据双强迫多项式的定义，同时也通过图的对比发现，当两个七个苯环组成的六角系统双强迫多项式相等时，它们的强迫多项式和反强迫多项式一定相等。但是反过来却未必成立。

Table 1. Di-Forced polynomials of cata-condensed hexagonal systems generated by seven benzene rings^*

表1. 七个苯环生成的cata-型六角系统的双强迫多项式^*

H	$Faf\left(H;x,y\right)$	$\Phi \left(H\right)$	IDF	ADF
H1	$2xy+6x{y}^2$	8	8	14
H2	$xy+3x^2{y}^2+9x^2{y}^3$	13	25	34
H3	$xy+3x^2{y}^2+9x^2{y}^3+3x^2{y}^4$	16	31	46
H4	$5x^2{y}^2+8x^2{y}^3+4x^2{y}^4$	17	34	50
H5	$5x^2{y}^2+8x^2{y}^3+4x^2{y}^4$	17	34	50
H6	$xy+3x^2{y}^2+9x^2{y}^3+4x^2{y}^4$	17	33	50
H7	$5x^2{y}^2+8x^2{y}^3+6x^2{y}^4$	19	38	58
H8	$5x^2{y}^2+8x^2{y}^3+6x^2{y}^4$	19	38	58
H9	$4x^2{y}^2+2x^2{y}^3+4x^3{y}^3+10x^3{y}^4+2x^3{y}^5$	22	60	76
H10	$4x^2{y}^2+2x^2{y}^3+4x^3{y}^3+10x^3{y}^4+2x^3{y}^5$	22	60	76
H11	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+8x^3{y}^4+3x^3{y}^5$	23	62	80
H12	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+8x^3{y}^4+3x^3{y}^5$	23	62	80
H13	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+8x^3{y}^4+3x^3{y}^5$	23	62	80
H14	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+8x^3{y}^4+3x^3{y}^5$	23	62	80
H15	$4x^2{y}^2+4x^3{y}^3+12x^3{y}^4$	20	56	68
H16	$4x^2{y}^2+4x^3{y}^3+12x^3{y}^4$	20	56	68
H17	$4x^2{y}^2+x^2{y}^3+4x^3{y}^3+11x^3{y}^4+3x^3{y}^5$	23	64	82
H18	$4x^2{y}^2+x^2{y}^3+4x^3{y}^3+11x^3{y}^4+3x^3{y}^5$	23	64	82
H19	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+9x^3{y}^4+5x^3{y}^5$	25	72	92
H20	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+9x^3{y}^4+5x^3{y}^5$	25	72	92
H21	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+9x^3{y}^4+5x^3{y}^5$	25	72	92
H22	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+9x^3{y}^4+5x^3{y}^5$	25	72	92
H23	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+8x^3{y}^4+5x^3{y}^5$	25	68	90
H24	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+8x^3{y}^4+5x^3{y}^5$	25	68	90
H25	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+8x^3{y}^4+5x^3{y}^5$	25	68	90
H26	$4x^2{y}^2+2x^2{y}^3+4x^3{y}^3+10x^3{y}^4+4x^3{y}^5$	24	66	86
H27	$4x^2{y}^2+2x^2{y}^3+4x^3{y}^3+10x^3{y}^4+4x^3{y}^5$	24	66	86
H28	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+10x^3{y}^4+5x^3{y}^5+x^3{y}^6$	27	78	102
H29	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+10x^3{y}^4+5x^3{y}^5+x^3{y}^6$	27	78	102
H30	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+10x^3{y}^4+5x^3{y}^5+x^3{y}^6$	27	78	102
H31	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+10x^3{y}^4+5x^3{y}^5+x^3{y}^6$	27	78	102
H32	$x^2{y}^2+10x^3{y}^3+12x^3{y}^4+4x^3{y}^5+2x^3{y}^6$	29	86	112
H33	$x^2{y}^2+10x^3{y}^3+12x^3{y}^4+4x^3{y}^5+2x^3{y}^6$	29	86	112
H34	$x^2{y}^2+10x^3{y}^3+12x^3{y}^4+4x^3{y}^5+2x^3{y}^6$	29	86	112
H35	$x^2{y}^2+10x^3{y}^3+12x^3{y}^4+4x^3{y}^5+2x^3{y}^6$	29	86	112
H36	$x^2{y}^2+10x^3{y}^3+12x^3{y}^4+4x^3{y}^5+2x^3{y}^6$	29	86	112
H37	$x^2{y}^2+10x^3{y}^3+12x^3{y}^4+4x^3{y}^5+2x^3{y}^6$	29	86	112
H38	$x^2{y}^2+10x^3{y}^3+2x^3{y}^4+5x^4{y}^4+10x^4{y}^5+x^4{y}^6$	29	102	116
H39	$x^2{y}^2+10x^3{y}^3+2x^3{y}^4+5x^4{y}^4+10x^4{y}^5+x^4{y}^6$	29	102	116
H40	$x^2{y}^2+10x^3{y}^3+2x^3{y}^4+5x^4{y}^4+10x^4{y}^5+x^4{y}^6$	29	102	116
H41	$2x^2{y}^2+2x^2{y}^3+8x^3{y}^3+8x^3{y}^4+6x^3{y}^5$	26	74	96
H42	$2x^2{y}^2+2x^2{y}^3+8x^3{y}^3+8x^3{y}^4+6x^3{y}^5$	26	74	96
H43	$2x^2{y}^2+2x^2{y}^3+8x^3{y}^3+8x^3{y}^4+6x^3{y}^5$	26	74	96
H44	$2x^2{y}^2+2x^2{y}^3+8x^3{y}^3+8x^3{y}^4+6x^3{y}^5$	26	74	96
H45	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H46	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H47	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H48	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H49	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H50	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H51	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H52	$x^2{y}^2+8x^3{y}^3+6x^3{y}^4+7x^4{y}^4+6x^4{y}^5+3x^4{y}^6$	31	108	126
H53	$12x^3{y}^3+10x^3{y}^4+6x^3{y}^5+2x^3{y}^6$	30	90	118
H54	$12x^3{y}^3+10x^3{y}^4+6x^3{y}^5+2x^3{y}^6$	30	90	118
H55	$12x^3{y}^3+10x^3{y}^4+6x^3{y}^5+2x^3{y}^6$	30	90	118
H56	$12x^3{y}^3+10x^3{y}^4+6x^3{y}^5+2x^3{y}^6$	30	90	118
H57	$12x^3{y}^3+10x^3{y}^4+6x^3{y}^5+2x^3{y}^6$	30	90	118
H58	$12x^3{y}^3+10x^3{y}^4+6x^3{y}^5+2x^3{y}^6$	30	90	118
H59	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H60	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H61	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H62	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H63	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H64	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H65	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H66	$9x^3{y}^3+6x^3{y}^4+8x^4{y}^4+3x^3{y}^5+5x^4{y}^5+2x^4{y}^6+x^4{y}^7$	34	118	142
H67	$x^2{y}^2+10x^3{y}^3+12x^3{y}^4+4x^3{y}^5+2x^3{y}^6$	29	86	112
H68	$x{y}^2+7x^3{y}^3+13x^3{y}^4$	21	61	75
H69	$x{y}^2+7x^3{y}^3+13x^3{y}^4+4x^3{y}^5$	25	73	95
H70	$x^2{y}^2+3x^2{y}^3+7x^3{y}^3+10x^3{y}^4+3x^3{y}^5$	24	68	87
H71	$x^2{y}^2+2x^2{y}^3+7x^3{y}^3+11x^3{y}^4+6x^3{y}^5$	27	78	103
H72	$x^2{y}^2+2x^2{y}^3+7x^3{y}^3+11x^3{y}^4+6x^3{y}^5$	27	78	103
H73	$x^2{y}^2+x^2{y}^3+7x^3{y}^3+12x^3{y}^4+5x^3{y}^5$	26	76	99
H74	$x^2{y}^2+x^2{y}^3+7x^3{y}^3+12x^3{y}^4+5x^3{y}^5$	26	76	99
H75	$2x^2{y}^3+4x^3{y}^3+10x^4{y}^4+14x^4{y}^5$	30	112	128
H76	$x^2{y}^2+x^2{y}^3+7x^3{y}^3+12x^3{y}^4+7x^3{y}^5+x^3{y}^6$	29	85	115
H77	$x{y}^2+7x^3{y}^3+13x^3{y}^4+6x^3{y}^5+x^3{y}^6$	28	82	111
H78	$x^2{y}^2+x^2{y}^3+3x^3{y}^3+2x^2{y}^4+x^3{y}^4+11x^4{y}^4+11x^4{y}^5+2x^4{y}^6$	32	116	137
H79	$x^2{y}^2+x^2{y}^3+3x^3{y}^3+2x^2{y}^4+x^3{y}^4+11x^4{y}^4+11x^4{y}^5+2x^4{y}^6$	32	116	137
H80	$10x^3{y}^3+13x^3{y}^4+5x^3{y}^5+2x^3{y}^6$	30	90	119
H81	$10x^3{y}^3+13x^3{y}^4+5x^3{y}^5+2x^3{y}^6$	30	90	119
H82	$10x^3{y}^3+13x^3{y}^4+5x^3{y}^5+2x^3{y}^6$	30	90	119
H83	$10x^3{y}^3+13x^3{y}^4+5x^3{y}^5+2x^3{y}^6$	30	90	119
H84	$6x^3{y}^3+4x^3{y}^4+9x^4{y}^4+12x^4{y}^5+3x^4{y}^6$	34	126	148
H85	$6x^3{y}^3+4x^3{y}^4+10x^4{y}^4+10x^4{y}^5+4x^4{y}^6$	34	126	148
H86	$6x^3{y}^3+4x^3{y}^4+10x^4{y}^4+10x^4{y}^5+4x^4{y}^6$	34	126	148
H87	$6x^3{y}^3+4x^3{y}^4+10x^4{y}^4+10x^4{y}^5+4x^4{y}^6$	34	126	148
H88	$6x^3{y}^3+4x^3{y}^4+10x^4{y}^4+10x^4{y}^5+4x^4{y}^6$	34	126	148
H89	$x^2{y}^2+x^2{y}^3+3x^3{y}^3+x^2{y}^4+3x^3{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$	33	120	142
H90	$x^2{y}^2+4x^3{y}^3+4x^3{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$	33	122	142
H91	$x^2{y}^2+4x^3{y}^3+4x^3{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$	33	122	142
H92	$x^2{y}^2+x^2{y}^3+3x^3{y}^3+x^2{y}^4+3x^3{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$	33	120	142
H93	$x^2{y}^2+x^2{y}^3+3x^3{y}^3+x^2{y}^4+3x^3{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$	33	120	142
H94	$x^2{y}^2+x^2{y}^3+3x^3{y}^3+x^2{y}^4+3x^3{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$	33	120	142
H95	$10x^3{y}^3+12x^3{y}^4+7x^3{y}^5+2x^3{y}^6$	31	93	125
H96	$10x^3{y}^3+12x^3{y}^4+7x^3{y}^5+2x^3{y}^6$	31	93	125
H97	$10x^3{y}^3+12x^3{y}^4+7x^3{y}^5+2x^3{y}^6$	31	93	125
H98	$2x^2{y}^3+4x^3{y}^3+2x^3{y}^4+10x^4{y}^4+12x^4{y}^5+2x^4{y}^6$	32	118	138
H99	$2x^2{y}^3+4x^3{y}^3+2x^3{y}^4+10x^4{y}^4+12x^4{y}^5+2x^4{y}^6$	32	118	138
H100	$2x^2{y}^3+4x^3{y}^3+x^2{y}^4+10x^4{y}^4+13x^4{y}^5+x^4{y}^6$	31	114	133
H101	$6x^3{y}^3+4x^3{y}^4+11x^4{y}^4+2x^3{y}^5+9x^4{y}^5+3x^4{y}^6+x^4{y}^7$	36	132	158
H102	$6x^3{y}^3+4x^3{y}^4+11x^4{y}^4+2x^3{y}^5+9x^4{y}^5+3x^4{y}^6+x^4{y}^7$	36	132	158
H103	$6x^3{y}^3+4x^3{y}^4+11x^4{y}^4+2x^3{y}^5+9x^4{y}^5+3x^4{y}^6+x^4{y}^7$	36	132	158
H104	$6x^3{y}^3+4x^3{y}^4+11x^4{y}^4+2x^3{y}^5+9x^4{y}^5+3x^4{y}^6+x^4{y}^7$	36	132	158
H105	$6x^3{y}^3+4x^3{y}^4+11x^4{y}^4+2x^3{y}^5+9x^4{y}^5+3x^4{y}^6+x^4{y}^7$	36	132	158
H106	$x^2{y}^3+6x^3{y}^4+2x^3{y}^5+25x^5{y}^5+6x^5{y}^6+x^5{y}^7$	41	186	205
H107	$6x^3{y}^3+4x^3{y}^4+11x^4{y}^4+2x^3{y}^5+9x^4{y}^5+3x^4{y}^6+x^4{y}^7$	36	132	158
H108	$6x^3{y}^3+4x^3{y}^4+11x^4{y}^4+2x^3{y}^5+9x^4{y}^5+3x^4{y}^6+x^4{y}^7$	36	132	158
H109	$2x^2{y}^3+x^2{y}^4+18x^4{y}^4+9x^4{y}^5+5x^4{y}^6$	35	134	157
H110	$2x^2{y}^3+x^2{y}^4+18x^4{y}^4+9x^4{y}^5+5x^4{y}^6$	35	134	157
H111	$x^2{y}^3+2x^3{y}^3+x^3{y}^4+17x^4{y}^4+x^3{y}^5+10x^4{y}^5+4x^4{y}^6+x^4{y}^7$	37	142	167
H112	$x^2{y}^3+2x^3{y}^3+x^3{y}^4+17x^4{y}^4+x^3{y}^5+10x^4{y}^5+4x^4{y}^6+x^4{y}^7$	37	142	167
H113	$x^2{y}^3+2x^3{y}^3+x^3{y}^4+17x^4{y}^4+x^3{y}^5+10x^4{y}^5+4x^4{y}^6+x^4{y}^7$	37	142	167
H114	$3x^3{y}^3+2x^3{y}^4+16x^4{y}^4+x^3{y}^5+10x^4{y}^5+5x^4{y}^6+x^4{y}^7$	38	146	173
H115	$3x^3{y}^3+2x^3{y}^4+16x^4{y}^4+x^3{y}^5+10x^4{y}^5+5x^4{y}^6+x^4{y}^7$	38	146	173
H116	$7x^3{y}^3+9x^3{y}^4+8x^4{y}^4+3x^3{y}^5+4x^4{y}^5+3x^4{y}^6+x^4{y}^7$	35	121	149
H117	$7x^3{y}^3+9x^3{y}^4+8x^4{y}^4+3x^3{y}^5+4x^4{y}^5+3x^4{y}^6+x^4{y}^7$	35	121	149
H118	$8x^3{y}^4+24x^5{y}^5+8x^5{y}^6$	40	184	200

*后3列数据取自文献[19]。

Table 2. Di-Forced polynomials of peri-type hexagonal system with perfect matchings seven benzene rings^*

表2. 七个苯环生成的有完美匹配的peri-型六角系统的双强迫多项式^*

H	$Faf\left(H;x,y\right)$	$\Phi \left(H\right)$	IDF	ADF
H119	$x^2{y}^2+6x^3{y}^3+2x^3{y}^4+9x^4{y}^4+6x^4{y}^5+x^4{y}^6$	25	90	100
H120	$x^2{y}^2+6x^3{y}^3+2x^3{y}^4+9x^4{y}^4+6x^4{y}^5+x^4{y}^6$	25	90	100
H121	$2x^2{y}^2+x^2{y}^3+6x^3{y}^3+5x^3{y}^4+x^3{y}^5$	15	42	50
H122	$2x^2{y}^2+x^2{y}^3+6x^3{y}^3+5x^3{y}^4+x^3{y}^5$	15	42	50
H123	$2x^2{y}^2+2x^2{y}^3+6x^3{y}^3+8x^3{y}^4+2x^3{y}^5$	20	56	70
H124	$2x^2{y}^2+2x^2{y}^3+6x^3{y}^3+8x^3{y}^4+2x^3{y}^5$	20	56	70
H125	$10x^3{y}^3+9x^3{y}^4+4x^3{y}^5+x^3{y}^6$	24	72	92
H126	$10x^3{y}^3+9x^3{y}^4+4x^3{y}^5+x^3{y}^6$	24	72	92
H127	$4x^2{y}^2+4x^2{y}^3+x^2{y}^4$	9	18	24
H128	$4x^2{y}^2+4x^2{y}^3+x^2{y}^4$	9	18	24
H129	$4x^2{y}^2+4x^2{y}^3+x^2{y}^4$	9	18	24
H130	$4x^2{y}^2+8x^2{y}^3+4x^2{y}^4$	16	32	48
H131	$4x^2{y}^2+8x^2{y}^3+4x^2{y}^4$	16	32	48
H132	$4x^2{y}^2+6x^2{y}^3+2x^2{y}^4$	12	24	34
H133	$4x^2{y}^2+6x^2{y}^3+2x^2{y}^4$	12	24	34
H134	$2x^2{y}^2+x^2{y}^3+6x^3{y}^3+9x^3{y}^4+3x^3{y}^5$	21	60	76
H135	$2x^2{y}^2+x^2{y}^3+6x^3{y}^3+9x^3{y}^4+3x^3{y}^5$	21	60	76
H136	$2x^2{y}^2+x^2{y}^3+6x^3{y}^3+9x^3{y}^4+3x^3{y}^5$	21	60	76
H137	$2x^2{y}^2+x^2{y}^3+6x^3{y}^3+9x^3{y}^4+3x^3{y}^5$	21	60	76
H138	$4x^2{y}^2+2x^2{y}^3+4x^3{y}^3+6x^3{y}^4+2x^3{y}^5$	18	48	60
H139	$4x^2{y}^2+8x^2{y}^3+3x^2{y}^4$	15	30	44
H140	$4x^2{y}^2+8x^2{y}^3+3x^2{y}^4$	15	30	44
H141	$x{y}^2+2x^2{y}^2+7x^3{y}^3+7x^3{y}^4+2x^3{y}^5$	19	53	65
H142	$x{y}^2+9x^3{y}^3+9x^3{y}^4+4x^3{y}^5$	23	67	85
H143	$2x^2{y}^3+2x^3{y}^3+2x^3{y}^4+12x^4{y}^4+10x^4{y}^5+2x^4{y}^6$	30	112	130
H144	$2x^2{y}^3+2x^3{y}^3+x^2{y}^4+2x^3{y}^4+12x^4{y}^4+9x^4{y}^5+3x^4{y}^6$	31	114	135
H145	$4x^2{y}^3+x^2{y}^4+12x^4{y}^4+11x^4{y}^5+x^4{y}^6$	29	106	125
H146	$2x^2{y}^2+x^2{y}^3+5x^3{y}^3+x^2{y}^4+3x^3{y}^4+8x^4{y}^4+6x^4{y}^5+2x^4{y}^6$	28	96	112
H147	$3x^2{y}^2+2x^2{y}^3+5x^3{y}^3+9x^3{y}^4+4x^3{y}^5$	23	64	83
H148	$3x^2{y}^2+4x^2{y}^3+5x^3{y}^3+7x^3{y}^4+4x^3{y}^5$	23	62	87
H149	$x^2{y}^2+8x^3{y}^3+4x^3{y}^4+6x^4{y}^4+8x^4{y}^5+2x^4{y}^6$	29	102	118
H150	$x{y}^2+9x^3{y}^3+10x^3{y}^4+4x^3{y}^5+x^3{y}^6$	25	73	95
H151	$2x^2{y}^2+x^2{y}^3+5x^3{y}^3+x^2{y}^4+3x^3{y}^4+8x^4{y}^4+6x^4{y}^5+2x^4{y}^6$	28	96	112
H152	$x^2{y}^2+x^2{y}^3+9x^3{y}^3+9x^3{y}^4+5x^3{y}^5+x^3{y}^6$	26	76	99
H153	$x{y}^2+2x^2{y}^2+2x^2{y}^3+5x^3{y}^3+9x^3{y}^4+2x^3{y}^5$	21	57	73
H154	$3x^2{y}^2+2x^2{y}^3+5x^3{y}^3+9x^3{y}^4+2x^3{y}^5$	21	58	73
H155	$x^2{y}^2+x^2{y}^3+9x^3{y}^3+9x^3{y}^4+5x^3{y}^5+x^3{y}^6$	26	76	99
H156	$2x^2{y}^2+8x^3{y}^3+10x^3{y}^4+3x^3{y}^5+x^3{y}^6$	24	70	89
H157	$x{y}^2+5x^3{y}^3+5x^3{y}^4+10x^4{y}^4+4x^4{y}^5+2x^4{y}^6$	27	95	109
H158	$x^2{y}^2+x^2{y}^3+9x^3{y}^3+8x^3{y}^4+5x^3{y}^5$	24	70	89
H159	$x^2{y}^2+2x^2{y}^3+6x^3{y}^3+2x^3{y}^4+7x^4{y}^4+8x^4{y}^5+x^4{y}^6$	27	94	108
H160	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+9x^3{y}^4+4x^3{y}^5+x^3{y}^6$	25	72	93
H161	$x^2{y}^2+x^2{y}^3+7x^3{y}^3+x^2{y}^4+x^3{y}^4+7x^4{y}^4+8x^4{y}^5+x^4{y}^6$	27	94	108
H162	$x^2{y}^2+x^2{y}^3+7x^3{y}^3+x^2{y}^4+x^3{y}^4+7x^4{y}^4+8x^4{y}^5+x^4{y}^6$	27	94	108
H163	$2x^2{y}^3+6x^3{y}^3+4x^3{y}^4+8x^4{y}^4+6x^4{y}^5+2x^4{y}^6$	28	98	114
H164	$2x^2{y}^3+6x^3{y}^3+4x^3{y}^4+8x^4{y}^4+6x^4{y}^5+2x^4{y}^6$	28	98	114
H165	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+9x^3{y}^4+4x^3{y}^5+x^3{y}^6$	25	72	93
H166	$2x^2{y}^2+x^2{y}^3+8x^3{y}^3+9x^3{y}^4+4x^3{y}^5+x^3{y}^6$	25	72	93
H167	$x^2{y}^2+x^2{y}^3+7x^3{y}^3+x^2{y}^4+x^3{y}^4+7x^4{y}^4+8x^4{y}^5+x^4{y}^6$	27	94	108
H168	$5x^2{y}^2+4x^2{y}^3+3x^3{y}^3+6x^3{y}^4+3x^3{y}^5$	21	54	70
H169	$6x^2{y}^2+6x^2{y}^3+5x^2{y}^4$	17	34	50
H170	$5x^2{y}^2+4x^2{y}^3+3x^3{y}^3+6x^3{y}^4+3x^3{y}^5$	21	54	70
H171	$3x^2{y}^2+x^2{y}^3+8x^3{y}^3+8x^3{y}^4+3x^3{y}^5+x^3{y}^6$	24	68	86
H172	$3x^2{y}^2+x^2{y}^3+8x^3{y}^3+8x^3{y}^4+3x^3{y}^5+x^3{y}^6$	24	68	86
H173	$3x^2{y}^2+x^2{y}^3+8x^3{y}^3+8x^3{y}^4+3x^3{y}^5+x^3{y}^6$	24	68	86
H174	$3x^2{y}^2+x^2{y}^3+8x^3{y}^3+8x^3{y}^4+3x^3{y}^5+x^3{y}^6$	24	68	86
H175	$4x^2{y}^2+x^2{y}^3+6x^3{y}^3+7x^3{y}^4+3x^3{y}^5$	21	58	72
H176	$3x^2{y}^2+3x^2{y}^3+7x^3{y}^3+5x^3{y}^4+4x^3{y}^5$	22	60	76
H177	$3x^2{y}^2+3x^2{y}^3+7x^3{y}^3+5x^3{y}^4+4x^3{y}^5$	22	60	76
H178	$3x^2{y}^2+8x^3{y}^3+9x^3{y}^4+2x^3{y}^5+x^3{y}^6$	23	66	82
H179	$3x^2{y}^2+8x^3{y}^3+9x^3{y}^4+2x^3{y}^5+x^3{y}^6$	23	66	82
H180	$4x^2{y}^2+x^2{y}^3+6x^3{y}^3+7x^3{y}^4+3x^3{y}^5$	21	58	72
H181	$xy+4x^2{y}^2+7x^2{y}^3+3x^2{y}^4$	15	29	42
H182	$xy+2x^2{y}^2+4x^2{y}^3+5x^3{y}^3+5x^3{y}^4+2x^3{y}^5$	19	49	62
H183	$xy+5x^2{y}^2+5x^2{y}^3+3x^2{y}^4$	14	27	38
H184	$xy+4x^2{y}^2+3x^2{y}^3+3x^3{y}^3+4x^3{y}^4+x^3{y}^5$	16	39	48
H185	$6x^2{y}^2+4x^2{y}^3+2x^3{y}^3+4x^3{y}^4+2x^3{y}^5$	18	44	56
H186	$x{y}^2+9x^3{y}^3+9x^3{y}^4+3x^3{y}^5+x^3{y}^6$	23	67	86
H187	$x{y}^2+x^2{y}^2+9x^3{y}^3+8x^3{y}^4+2x^3{y}^5+x^3{y}^6$	22	63	79
H188	$x^2{y}^2+3x^2{y}^3+3x^3{y}^3+x^2{y}^4+x^3{y}^4+9x^4{y}^4+6x^4{y}^5+x^4{y}^6$	25	86	100
H189	$4x^2{y}^2+4x^2{y}^3+4x^3{y}^3+5x^3{y}^4+3x^3{y}^5$	20	52	67
H190	$5x^2{y}^2+6x^3{y}^3+8x^3{y}^4+x^3{y}^6$	20	55	88

*后3列数据取自文献[19]。

Figure 1. Three polynomial diagrams of the hexagonal system generated by seven benzene rings

图1. 七个苯环生成的六角系统三种多项式关系图

观察1 当两个由七个苯环组成的六角系统的强迫多项式相等时，其双强迫多项式不一定相等。

$F\left(H_{28},x\right)=3x^2+24x^3=F\left(H_{71},x\right)$

$Faf\left(H_{28};x,y\right)=2x^2{y}^2+x^2{y}^3+8x^3{y}^3+10x^3{y}^4+5x^3{y}^5+x^3{y}^6$

$Faf\left(H_{71};x,y\right)=x^2{y}^2+2x^2{y}^3+7x^3{y}^3+11x^3{y}^4+6x^3{y}^5$

H₂₈和H₇₁强迫多项式相同，但二者双强迫多项式不同。

观察2 当两个由七个苯环组成的六角系统的反强迫多项式相等时，双强迫多项式也不一定相等。

$Af\left(H_{89},x\right)=x^2+4x^3+15x^4+10x^5+3x^6=Af\left(H_{90},x\right)$

$Faf\left(H_{89};x,y\right)=x^2{y}^2+x^2{y}^3+3x^3{y}^3+x^2{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$

$Faf\left(H_{90};x,y\right)=x^2{y}^2+4x^3{y}^3+4x^3{y}^4+11x^4{y}^4+10x^4{y}^5+3x^4{y}^6$

H₈₉和H₉₀反强迫多项式相同，双强迫多项式不同。

还存在一种更加特殊的情况。

观察3 当两个由七个苯环组成的六角系统的强迫多项式与反强迫多项式都相等时，双强迫多项式也不一定相等。

$F\left(H_{159},x\right)=3x^2+8x^3+16x^4=F\left(H_{161},x\right)$

$Af\left(H_{159},x\right)=x^2+8x^3+9x^4+8x^5+x^6=Af\left(H_{161},x\right)$

$Faf\left(H_{159};x,y\right)=x^2{y}^2+2x^2{y}^3+6x^3{y}^3+2x^3{y}^4+7x^4{y}^4+8x^4{y}^5+x^4{y}^6$

$Faf\left(H_{161};x,y\right)=x^2{y}^2+x^2{y}^3+7x^3{y}^3+x^2{y}^4+x^3{y}^4+7x^4{y}^4+8x^4{y}^5+x^4{y}^6$

H₁₆₁与H₁₅₉的强迫多项式和反强迫多项式都相等，但二者的双强迫多项式不相等。由此可得出双强迫多项式虽然计算较为复杂，但相比于强迫多项式和反强迫多项式，具有更好的区分性，根据双强迫多项式能够更加精确的划分六角系统。除了双强迫多项式、强迫多项式、反强迫多项式，还可以考虑另外一些指标的区分情况。

下面以图表形式展现双强迫多项式，强迫多项式，反强迫多项式，完美匹配个数，自由度以及反自由度对由七个苯环组成的六角系统图的区分情况。可以用该性质对图进行区分的打Π，无法利用该性质对图区分打Ο。

Table 3. Differentiation of a perfectly matched hexagonal system generated by seven benzene rings^*

表3. 对七个苯环生成的有完美匹配的六角系统的区分情况^*

H	$Faf\left(H;x,y\right)$	$F\left(H,x\right)$	$Af\left(H,x\right)$	$\Phi \left(H\right)$	IDF	ADF
H1	Π	Π	Π	Π	Π	Π
H2	Π	Π	Π	Π	Π	Ο
H3	Π	Π	Π	Ο	Π	Π
H4	Ο	Ο	Ο	Ο	Ο	Ο
H5	Ο	Ο	Ο	Ο	Ο	Ο
H6	Π	Π	Π	Ο	Π	Ο
H7	Ο	Ο	Ο	Ο	Ο	Ο
H8	Ο	Ο	Ο	Ο	Ο	Ο
H9	Ο	Ο	Ο	Ο	Ο	Ο
H10	Ο	Ο	Ο	Ο	Ο	Ο
H11	Ο	Ο	Ο	Ο	Ο	Ο
H12	Ο	Ο	Ο	Ο	Ο	Ο
H13	Ο	Ο	Ο	Ο	Ο	Ο
H14	Ο	Ο	Ο	Ο	Ο	Ο
H15	Ο	Ο	Ο	Ο	Ο	Ο
H16	Ο	Ο	Ο	Ο	Ο	Ο
H17	Ο	Ο	Ο	Ο	Ο	Ο
H18	Ο	Ο	Ο	Ο	Ο	Ο
H19	Ο	Ο	Ο	Ο	Ο	Ο
H20	Ο	Ο	Ο	Ο	Ο	Ο
H21	Ο	Ο	Ο	Ο	Ο	Ο
H22	Ο	Ο	Ο	Ο	Ο	Ο
H23	Ο	Ο	Ο	Ο	Ο	Ο
H24	Ο	Ο	Ο	Ο	Ο	Ο
H25	Ο	Ο	Ο	Ο	Ο	Ο
H26	Ο	Ο	Ο	Ο	Ο	Ο
H27	Ο	Ο	Ο	Ο	Ο	Ο
H28	Ο	Ο	Ο	Ο	Ο	Ο
H29	Ο	Ο	Ο	Ο	Ο	Ο
H30	Ο	Ο	Ο	Ο	Ο	Ο
H31	Ο	Ο	Ο	Ο	Ο	Ο
H32	Ο	Ο	Ο	Ο	Ο	Ο
H33	Ο	Ο	Ο	Ο	Ο	Ο
H34	Ο	Ο	Ο	Ο	Ο	Ο
H35	Ο	Ο	Ο	Ο	Ο	Ο
H36	Ο	Ο	Ο	Ο	Ο	Ο
H37	Ο	Ο	Ο	Ο	Ο	Ο
H38	Ο	Ο	Ο	Ο	Ο	Ο
H39	Ο	Ο	Ο	Ο	Ο	Ο
H40	Ο	Ο	Ο	Ο	Ο	Ο
H41	Ο	Ο	Ο	Ο	Ο	Ο
H42	Ο	Ο	Ο	Ο	Ο	Ο
H43	Ο	Ο	Ο	Ο	Ο	Ο
H44	Ο	Ο	Ο	Ο	Ο	Ο
H45	Ο	Ο	Ο	Ο	Ο	Ο
H46	Ο	Ο	Ο	Ο	Ο	Ο
H47	Ο	Ο	Ο	Ο	Ο	Ο
H48	Ο	Ο	Ο	Ο	Ο	Ο
H49	Ο	Ο	Ο	Ο	Ο	Ο
H50	Ο	Ο	Ο	Ο	Ο	Ο
H51	Ο	Ο	Ο	Ο	Ο	Ο
H52	Ο	Ο	Ο	Ο	Ο	Ο
H53	Ο	Ο	Ο	Ο	Ο	Ο
H54	Ο	Ο	Ο	Ο	Ο	Ο
H55	Ο	Ο	Ο	Ο	Ο	Ο
H56	Ο	Ο	Ο	Ο	Ο	Ο
H57	Ο	Ο	Ο	Ο	Ο	Ο
H58	Ο	Ο	Ο	Ο	Ο	Ο
H59	Ο	Ο	Ο	Ο	Ο	Ο
H60	Ο	Ο	Ο	Ο	Ο	Ο
H61	Ο	Ο	Ο	Ο	Ο	Ο
H62	Ο	Ο	Ο	Ο	Ο	Ο
H63	Ο	Ο	Ο	Ο	Ο	Ο
H64	Ο	Ο	Ο	Ο	Ο	Ο
H65	Ο	Ο	Ο	Ο	Ο	Ο
H66	Ο	Ο	Ο	Ο	Ο	Ο
H67	Ο	Ο		Ο	Ο	Ο
H68	Π	Π	Π	Ο	Π	Π
H69	Π	Π	Π	Ο	Ο	Ο
H70	Π	Π	Π	Ο	Ο	Ο
H71	Ο	Ο	Ο	Ο	Ο	Ο
H72	Ο	Ο	Ο	Ο	Ο	Ο
H73	Ο	Ο	Ο	Ο	Ο	Ο
H74	Ο	Ο	Ο	Ο	Ο	Ο
H75	Π	Π	Π	Ο	Ο	Π
H76	Π	Π	Π	Ο	Π	Π
H77	Π	Π	Π	Ο	Π	Π
H78	Ο	Ο	Ο	Ο	Ο	Ο
H79	Ο	Ο	Ο	Ο	Ο	Ο
H80	Ο	Ο	Ο	Ο	Ο	Ο
H81	Ο	Ο	Ο	Ο	Ο	Ο
H82	Ο	Ο	Ο	Ο	Ο	Ο
H83	Ο	Ο	Ο	Ο	Ο	Ο
H84	Π	Ο	Π	Ο	Ο	Ο
H85	Ο	Ο	Ο	Ο	Ο	Ο
H86	Ο	Ο	Ο	Ο	Ο	Ο
H87	Ο	Ο	Ο	Ο	Ο	Ο
H88	Ο	Ο	Ο	Ο	Ο	Ο
H89	Ο	Ο	Ο	Ο	Ο	Ο
H90	Ο	Ο	Ο	Ο	Ο	Ο
H91	Ο	Ο	Ο	Ο	Ο	Ο
H92	Ο	Ο	Ο	Ο	Ο	Ο
H93	Ο	Ο	Ο	Ο	Ο	Ο
H94	Ο	Ο	Ο	Ο	Ο	Ο
H95	Ο	Ο	Ο	Ο	Ο	Ο
H96	Ο	Ο	Ο	Ο	Ο	Ο
H97	Ο	Ο	Ο	Ο	Ο	Ο
H98	Ο	Ο	Ο	Ο	Ο	Ο
H99	Ο	Ο	Ο	Ο	Ο	Ο
H100	Π	Π	Π	Ο	Ο	Π
H101	Ο	Ο	Ο	Ο	Ο	Ο
H102	Ο	Ο	Ο	Ο	Ο	Ο
H103	Ο	Ο	Ο	Ο	Ο	Ο
H104	Ο	Ο	Ο	Ο	Ο	Ο
H105	Ο	Ο	Ο	Ο	Ο	Ο
H106	Π	Π	Π	Π	Π	Π
H107	Ο	Ο	Ο	Ο	Ο	Ο
H108	Ο	Ο	Ο	Ο	Ο	Ο
H109	Ο	Ο	Ο	Ο	Ο	Ο
H110	Ο	Ο	Ο	Ο	Ο	Ο
H111	Ο	Ο	Ο	Ο	Ο	Ο
H112	Ο	Ο	Ο	Ο	Ο	Ο
H113	Ο	Ο	Ο	Ο	Ο	Ο
H114	Ο	Ο	Ο	Ο	Ο	Ο
H115	Ο	Ο	Ο	Ο	Ο	Ο
H116	Ο	Ο	Ο	Ο	Ο	Ο
H117	Ο	Ο	Ο	Ο	Ο	Ο
H118	Π	Π	Π	Π	Π	Π
H119	Ο	Ο	Ο	Ο	Ο	Ο
H120	Ο	Ο	Ο	Ο	Ο	Ο
H121	Ο	Ο	Ο	Ο	Ο	Ο
H122	Ο	Ο	Ο	Ο	Ο	Ο
H123	Ο	Ο	Ο	Ο	Ο	Ο
H124	Ο	Ο	Ο	Ο	Ο	Ο
H125	Ο	Ο	Ο	Ο	Ο	Ο
H126	Ο	Ο	Ο	Ο	Ο	Ο
H127	Ο	Ο	Ο	Ο	Ο	Ο
H128	Ο	Ο	Ο	Ο	Ο	Ο
H129	Ο	Ο	Ο	Ο	Ο	Ο
H130	Ο	Ο	Ο	Ο	Ο	Ο
H131	Ο	Ο	Ο	Ο	Ο	Ο
H132	Ο	Ο	Ο	Ο	Ο	Ο
H133	Ο	Ο	Ο	Ο	Ο	Ο
H134	Ο	Ο	Ο	Ο	Ο	Ο
H135	Ο	Ο	Ο	Ο	Ο	Ο
H136	Π	Ο	Ο	Ο	Ο	Ο
H137	Ο	Ο	Ο	Ο	Ο	Ο
H138	Ο	Π	Π	Ο	Π	Π
H139	Π	Ο	Ο	Ο	Ο	Ο
H140	Π	Ο	Ο	Ο	Ο	Ο
H141	Π	Π	Π	Ο	Π	Π
H142	Π	Π	Π	Ο	Ο	Π
H143	Π	Π	Π	Ο	Ο	Π
H144	Ο	Π	Π	Ο	Ο	Π
H145	Π	Π	Π	Ο	Π	Ο
H146	Π	Ο	Ο	Ο	Ο	Ο
H147	Π	Π	Π	Ο	Ο	Π
H148	Π	Π	Π	Ο	Ο	Ο
H149	Ο	Π	Π	Ο	Ο	Ο
H150	Ο	Π	Π	Ο	Ο	Ο
H151	Π	Ο	Ο	Ο	Ο	Ο
H152	Π	Ο	Ο	Ο	Ο	Ο
H153	Ο	Π	Ο	Ο	Π	Ο
H154	Π	Ο	Ο	Ο	Ο	Ο
H155	Π	Ο	Ο	Ο	Ο	Ο
H156	Π	Ο	Π	Ο	Ο	Ο
H157	Π	Π	Π	Ο	Π	Π
H158	Ο	Ο	Π	Ο	Ο	Ο
H159	Ο	Ο	Ο	Ο	Ο	Ο
H160	Ο	Ο	Ο	Ο	Ο	Ο
H161	Ο	Ο	Ο	Ο	Ο	Ο
H162	Ο	Ο	Ο	Ο	Ο	Ο
H163	Ο	Ο	Ο	Ο	Ο	Ο
H164	Ο	Ο	Ο	Ο	Ο	Ο
H165	Ο	Ο	Ο	Ο	Ο	Ο
H166	Ο	Ο	Ο	Ο	Ο	Ο
H167	Π	Ο	Ο	Ο	Ο	Ο
H168	Ο	Ο	Ο	Ο	Ο	Ο
H169	Ο	Π	Π	Ο	Ο	Ο
H170	Ο	Ο	Ο	Ο	Ο	Ο
H171	Ο	Ο	Ο	Ο	Ο	Ο
H172	Ο	Ο	Ο	Ο	Ο	Ο
H173	Ο	Ο	Ο	Ο	Ο	Ο
H174	Ο	Ο	Ο	Ο	Ο	Ο
H175	Ο	Ο	Ο	Ο	Ο	Ο
H176	Ο	Ο	Ο	Ο	Ο	Ο
H177	Ο	Ο	Ο	Ο	Ο	Ο
H178	Π	Ο	Ο	Ο	Ο	Ο
H179	Π	Ο	Ο	Ο	Ο	Ο
H180	Π	Ο	Ο	Ο	Ο	Ο
H181	Π	Π	Π	Ο	Π	Π
H182	Π	Π	Π	Ο	Π	Π
H183	Π	Π	Π	Π	Π	Π
H184	Π	Π	Π	Ο	Π	Ο
H185	Π	Π	Π	Ο	Π	Π
H186	Π	Π	Π	Ο	Ο	Ο
H187	Π	Π	Π	Ο	Π	Π
H188	Π	Π	Π	Ο	Ο	Ο
H189	Ο	Π	Π	Ο	Π	Π
H190	Ο	Π	Π	Ο	Π	Π

*后5列数据取自文献[19]。

由表3可以看出，在由七个苯环生成的六角系统中，六种图的性质对于图的区分情况有明显差别。

观察4 按照区分度排序，双强迫多项式 > 反强迫多项式 > 强迫多项式 >> 反自由度 > 自由度 >> 完美匹配个数。

看出双强迫多项式可被区分图的数量最多，说明双强迫多项式对图的区分度最高。同时由可被区分百分比可以看出，双强迫多项式、强迫多项式和反强迫多项式三者区分百分比较为接近，且可区分的图较多。自由度与反自由度的区分百分比也较为接近对图的区分效果一般。完美匹配个数区分百分比较低，仅能对个别图进行区分。

同时，由可被区分类数可以发现190个由七个苯环生成的六角系统有种类最多的双强迫多项式，强迫多项式、反强迫多项式、自由度和反自由度类数适中，完美匹配个数的类数最少。

基金项目

国家自然科学基金(12161081)。