关于图的匹配强迫数和强迫谱的研究

In chemistry,a CC-double bond is called a perfect matching, which is the framework of carbon of the molecule of hytrocarbon,the matching forcing number of a graph is the minimum number of matching edges which can uniquely determine a perfect matching of the graph. Chemists found that the perfect matching with bigger forcing number contributes more to the stability of molecule. Therefore, it makes sense to study the forcing number for the study of stability of molecule.. Since the concept of forcing number is proposed in 1991, there are a lot of outcomes about the minimum forcing number of bipartite graphs, but there is little study about the maximum forcing number or forcing spectrum. The main contents of this study: discuss the maximum forcing number and the continuity of forcing spectrum of fullerene graphs and boron-notrogen fullerene graphs, respectively; characterize the boron-nitrogen fullerene graphs whose forcing number is 3; get the maximum forcing number of graphs on the surfaces with small genus by deleting an independent set; apply the connectivity of Z-transformation graph of plane elementary bipartite graphs to study the forcing spectrum of hexagonal system; consummate the outcomes of forcing number of special bipartite graphs. The meaning of this study is further improving the system theory of forcing number, and supply theory basis for the study of stability of molecule in chemistry.

在化学上，苯类碳氢化合物分子的碳原子骨架图中的一组CC-双键被称为此图的一个完美匹配，图的匹配强迫数就是能唯一确定一个图的完美匹配的最少匹配边数目。化学家们发现强迫数越大的完美匹配对分子稳定性的贡献越大，反之越小。因而研究图的匹配强迫数对于研究分子的稳定性有很大实际意义。. 自1991年图的匹配强迫数概念提出以来，已在二部图的最小匹配强迫数方面取得了较多研究成果，但在最大强迫数和强迫谱的研究方面进展甚小。本项目的主要研究内容：探讨富勒烯图，硼氮富勒烯图的最大强迫数及强迫谱的连续性；刻画硼氮富勒烯图强迫数为3 的图类；通过去独立点的方法求小亏格曲面上图的最大强迫数；运用平面基本二部图z-变换图的连通性研究六角系统的强迫谱；完善典型二部图强迫数的结论。本研究的意义是进一步完善强迫数的理论体系，为化学分子稳定性的研究提供一定的理论依据。

本项目研究了偶多边形链的匹配强迫谱，得到所有偶多边形链的强迫谱都是连续的；研究了硼氮富勒烯图的反强迫数，得出其反强迫数的下界是2，并刻画出所有达到下界的图；研究了k个边不交圈图的Kirchhoff指标，得到其指标的最小值，并刻画出极值图；研究了环面，克莱因瓶上四六八格图的强迫数，得到其最大强迫数。