图的匹配强迫、反强迫和整体强迫及其应用

This project mainly studies matching forcing, anti-forcing and entirely forcing problems of graphs, including forcing numbers and anti-forcing numbers of perfect matchings of graphs and their distributions, complete forcing numbers and relationships among various forcing invariants as well as their applications to molecular stability. In details, we shall discuss relations between the maximum forcing number and the resonance number and between maximum anti-forcing number and Fries number in plane bipartite graphs with perfect matchings, and the existence of a forced resonance set and the estimation of its size; We investigate forcing number distribution law of perfect matchings of graphs, and obtain the asymptotic behavior of the average forcing number of perfect matchings of some graphs by calculating their forcing polynomials; We also investigate some distribution laws of anti-forcing numbers of perfect matchings of graphs, and obtain the anti-forcing spectrum of plane and toroidal lattice graphs and some characteristics of anti-forcing polynomials of graphs; We shall compute the complete forcing numbers of some typical types of graphs and reveal some relationships between it with the global forcing number and the resonance number, and relations among various matching forcing invariants. Such forcing problems of graphs originate from theoretical chemistry. The implementation of this project can strengthen theoretical work on matching forcing of graphs that we have carried out, promote the further development of the matching theory of graphs. Meantime, a large number of forcing invariants will be applied to fullerenes, novel carbon molecules with polyhedral structures, so that to screen out suitable invariants of a good correlation with the stability.

本项目主要研究图的匹配强迫、反强迫和整体强迫问题，包括图的完美匹配的强迫数和反强迫数及其分布、完全强迫数、强迫不变量之间的关系及其在分子稳定性方面的应用。具体地讨论平面二部图中完美匹配的最大强迫数与共振数，最大反强迫数与Fries数的关系，强迫共振集的存在性及其大小的估计；研究图的完美匹配的强迫数分布规律，通过计算若干图的匹配强迫多项式得出平均强迫数的渐近行为；研究图的完美匹配的反强迫数分布规律，得出平面和环面格子图的反强迫谱，及图的反强迫多项式的特征；计算若干典型图的完全强迫数、揭示它与全局强迫数与共振数的关系，及其它匹配强迫不变量之关系。图的匹配各种强迫问题来源于理论化学，通过本项目的研究进一步深化我们已开展的图的匹配强迫的理论工作，推动匹配理论的进一步发展，同时将已出现的大量匹配强迫不变量应用于富勒烯这样的具有多面体结构的新型分子，筛选出与其稳定性有较好相关性的不变量。

该项目主要研究图的完美匹配的强迫与反强迫，完全强迫与全局强迫及其关系，源于理论化学中有机分子的克库勒结构及其内自由度，是图的匹配理论发展的新方向。在我们原有研究工作的基础上，提出了基本割分解等方法证明了连通图的完全强迫数以基圈数的2倍为非平凡上界并刻画了极值图，计算出了一些典型六角系统、轮图、柱面格子图、完全图和完全二部图及一些去边子图的完全强迫数公式，提出了(4,6)-富勒烯图的完全强迫数与Clar 数与Fries数之和相等的猜想；发现了全局强迫数与最大反强迫数之间的有趣关系，对二部图及更一般的Birkhoff-von Neuman图证实了全局强迫数大于等于最大反强迫数，并与传统匹配理论联系起来(如完美匹配多面体和solid brick等)，针对富勒烯图表明一定存在nice不交五边形对，解决了T. Doslic的四个公开问题；得到了凸六角系统的最小(反)强迫数，并证明了其强迫谱连续，对更一般的三可分单调六角形表明其强迫谱不一定连续，提出强迫谱最多有两个缺失值的猜想，表明了奇环面方格子图的共振图由同构的两个分支构成由此表明其强迫谱连续，表明凸六角系统的反强迫谱不一定连续，确定了两类管状(4,6)-富勒烯的反强迫谱的整区间表达；计算出了单层和双层直链方格子图的强迫和反强迫多项式，可构造单调六角系统以及具有一个转折的可构造六角系统的强迫多项式。完整解决了Che和Chen 于2011年提出刻画最小强迫数达到最大的图类的问题，通过匹配二交换方法证明了最大强迫数等于n-1的图的强迫谱是连续的，用点数和边数给出了图的最小强迫数的一般下界，从一个新的角度推广了国际著名学者G. Hetyei与L. Lovász的经典结果：具有2n个顶点且有唯一完美匹配的图最多有n^2条边。研究了六角系统的强迫共振数，给出了一般六角链的强迫共振数的算法，得到了一些具有递归结构的链状六角系统的强迫共振数的公式，比如zigzag链、双zigzag链和芘(pyrene)链等。