{(a,b),4}-球图共振性问题的研究

Since the resonant theory can predict the molecular stability in chemistry, the study of resonant problems for graphs is of great meaningful. Deza et al. firstly proposed the concept of {(a, b),4}-spheres, and they proved that if it exists {(a, b),4}-spheres, then a=2, b=4 or a=3, b=4, which corresponds to the {(2,4),4} and {(3,4),4}-spheres. This project will discuss the k-resonance of these two graphs, and describe all maximally resonant {(a, b),4}-spheres. Be different from the previous study on the resonant problems only for 3-regular graphs or the graphs whose inner vertices are of degree 3, this project researches the plane 4-regular graphs, which is a feature and innovation in this project. For a plane 3-regular graph, we can judge it is not 3-resonant by means of removing three faces incident to a vertex, and then isolating the vertex. But {(a, b),4}-spheres are plane 4-regular graphs, so how to study the 3-resonance of plane 4-regular graphs is the key problem for this project to solve. On the way, with the help of the characterization of Deza for {(a, b),4}-spheres and strengthen Tutte’s 1-factor theorem, combining with the reduction to absurdity, then we can obtain the desired results. Through the study of this project, on the one hand, the basic approaches and results on resonant problems of plane 4-regular graphs can be riched, on the other hand, it can provide theoretical guidance for later studying the stability of the corresponding molecule.

由于共振理论在化学中可以预测分子稳定性，因此对图的共振性问题的研究是有意义的。Deza 等人首先提出{(a,b),4}-球图的概念，并证明这样的图如果存在，则 a=2,b=4 或 a=3,b=4，即对应于{(2,4),4}和{(3,4),4}-球图。本项目将探讨这两类图的k-共振性问题，并刻画所有极大共振的{(a,b),4}-球图。对于平面3-正则图形，我们可以通过去掉与一个点关联的3 个面后孤立出该点，来判断该图不是3-共振的，而{(a,b),4}-球图是平面4-正则图形，那么如何研究平面4-正则图形的3-共振问题是本项目需要解决的关键问题。在方法上，借助于Deza 等人对这两类图性质的刻画，以及加强的Tutte 1-因子定理，结合反证法，从而得到结果。通过本项目的研究，一方面丰富平面4-正则图形共振性问题上的基本方法和基本结论，另一方面为以后研究这些图形所对应分子的稳定性提供理论指导。

分子稳定性是化学家们关注的一个重点内容，预测分子稳定性的指标有很多，其中一个就是共振理论。鉴于平面4-8格子图和{(a,b),m}-球图在化学图论中的重要作用，因此对这两类图共振性问题的研究是有意义的。本项目首先研究了两类平面4-8格子图G(p,q)和H(p,q)的共振性问题，证明了平面4-8格子图G(p,q)是极大共振的，H(p,q)是1,2-共振的，并且仅有的3-共振H(p,q)图有三个：H(0,q)，H(1,2)和H(2,2)，它们也是极大共振的，由此可以得到：H(p,q)是极大共振的当且仅当它是3-共振的。这一结论和以前研究过的苯系统，带洞六角系统，开口碳纳米管，B-N富勒烯，多边形系统，三正则二部多面体图，富勒烯，(3,6)-富勒烯图等图类共振性结论相吻合。其次我们研究了{(3,4),4}-球图的共振性问题，证明了每个{(3,4),4}-球图是1-共振的，并且{(3,4),4}-球图是极大共振的当且仅当它同构于12个图中的一个。由此得到一个推论：{(3,4),4}-球图是极大共振的当且仅当它是4－共振的。