七个苯环生成的六角系统的双强迫多项式

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。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.

苯环数目不超过六的六角系统的双强迫多项式

六角系统是一个2-连通的有限平面二部图,其中每个内面边界都是单位的正六边形。具有凯库勒结构的六角系统H的双强迫多项式是H的所有完美匹配的强迫数和反强迫数的二元计数多项式。本文计算了苯环数目不超过六的六角系统的双强迫多项式,由此得到其强迫多项式,反强迫多项式,内自由度与外自由度,为六角系统的结构分析提供了新的数学工具和结果。The hexagonal system is a 2-connected finite plane bipartite graph, in which each inner boundary is a regular hexagon of a unit. The di-forcing polynomials of hexagonal systems H with Kekulé structure are the binary counting polynomials of all perfect matchings forcing and anti-forcing numbers of H. In this paper, the di-forcing polynomials of hexagonal systems with no more than six benzene rings are calculated, from which the forcing polynomials, anti-forcing polynomials, internal and external degrees of freedom are obtained, it provides a new mathematical tool and results for the structural analysis of hexagonal system.

梯子图双强迫多项式的递推求解

梯子图Ln是路Pn和路P2的笛卡尔积。图的双强迫多项式是Ln的所有完美匹配的强迫数和反强迫数的二元计数多项式。通过对给定顶点关联边的匹配情况的分类讨论和计数,得出了梯子图双强迫多项式的递推公式,并由此计算出了其生成函数和一些低阶梯子图的双强迫多项式。