(k,6)-fullerene图与超立方图的匹配强迫和反强迫问题

The matching forcing and anti-forcing numbers of graphs as topology invariants are important for predicting the stability of molecular, which arouse enormous interest in mathematics and theoretical chemistry. Fullerene graphs are active fields in chemical graph theory, and its stability is always a significant theoretical topic. In this project, we will study the non-trivial upper bounds of the maximum forcing numbers of (3,6)-fullerenes and (5,6)-fullerenes, and characterize (4,6)-fullerenes with the forcing spectrum discontinuous. Moreover, the continuity of the anti-forcing spectrum of (4,6)-fullerenes will also be considered. Hypercubes are the most effective topology of the Internet. Its research dabble in various fields. Following the characterization of the maximum forcing number of hypercubes by Alon, its maximum anti-forcing number is also obtained. This project will also study the minimum anti-forcing number and the anti-forcing spectrum of hypercubes. The implementation of this project can promote the research of the stability of the Fullerenes, strengthen theoretical work on matching forcing and anti-forcing of hypercubes, and promote the further development of the matching theory of graphs.

图的匹配强迫和反强迫数作为拓扑不变量对预测分子的稳定性有重要意义，在数学和理论化学上引起了极大兴趣。Fullerene图是化学图论的活跃领域，其稳定性的研究一直是一个重要的理论课题。本项目将研究(3,6)-fullerene图和(5,6)-fullerene图的最大强迫数的非平凡上界；探索强迫谱不连续的(4,6)-fullerene图类的刻画；同时研究(4,6)-fullerene图反强迫谱的连续性。另外，作为最有效的互联网络拓扑结构，超立方图的研究涉猎各个领域。继著名图论专家Alon对超立方图的最大强迫数的研究之后，超立方图的最大反强迫数也已得到解决。本项目的另一项工作是求超立方图的最小反强迫数，并研究其匹配反强迫谱。这些工作将促进Fullerene稳定性的研究，深化超立方图的匹配强迫和反强迫的理论，从而进一步地推动匹配理论的发展。

本项目前期主要研究强迫谱不连续的(4,6)-富勒烯图类的刻画，以及(4,6)-富勒烯图的反强迫谱；探索(3,6)-富勒烯图和(5,6)-富勒烯图的最小强迫数刻画，及最大强迫数的上界；讨论超立方图的最小反强迫数及匹配反强迫谱。取得的主要结果有：得到了(4,6)-富勒烯图最大相容共振六边形面集大小的计算公式，给出了Tn和Bn两类(4,6)-富勒烯图的反强迫谱；刻画了最小强迫数等于3的富勒烯图；计算出了Q5的反强迫谱，从而发现超立方图的反强迫谱既不连续也不是等差数列。在坚持考虑以上重点问题的同时，后期本人与合作者研究了富勒烯图的完美星Packing，六角链和树的极大匹配以及Tough图的哈密尔顿性。