两类极值能量的问题的研究

Graph energy is a mathematical concept based on the approximation of energy of chemical molecule graph, which is a very active field on graph theory. Characterizing the graph having extremal value energy is an important aspect of the study of graph energy. So far, the major methods of study on extremal value are quasi-order and operation of graphs. Since the rapid development of these studies, a lot of problems which are incomparable by quasi-order have been appeared. Solutions for these problems will be to promote more better the theory study on graph energy. Meanwhile,these also provided theoretically base for prognosticating and composing with new compounds and materia medica. With our forepart work, utilizing the iteration of combinatorics and together with the knowledge on real analysis, the project will be tried to consider the kind problems and find a new method for solving the kind problems. What is more, we will try our best to research the bicyclic graph extremal energy conjecture proposed by Gutman et al. Furthermore, there are much close relation between graph matching energy and graph energy. The project will be considered to study the bound and extremal value of graph matching energy. By our research on matching energy, we will create condition for considering the problem that matching energy and energy have which relation proposed by Gutman et al.

图的能量是基于化学分子图的能量的近似而提出的数学概念；是图论的一个活跃的研究方向。刻画具有极值能量的图的结构是图能量研究的一个重要方面。目前研究极值能量的方法主要是拟序比较及图的运算。随着研究的进一步深入，出现了大量拟序不可比问题。这些拟序不可比问题的解决将会进一步完善图能量的理论研究，并为预测、合成新的化合物和新的药品方面提供理论基础。在前期工作的基础上，本项目将利用组合迭代法并结合实分析的知识去考虑这类问题，从而找到解决该类问题的新方法，力争彻底解决Gutman 等人提出的极大双圈图猜想。此外，图的匹配能量和图的能量有着密切联系。本项目将考虑图的匹配能量界及极值问题，并为研究Gutman等人提出关于图的匹配能量与图能量二者之间的确切关系的问题创造条件。

图的能量定义为图的特征多项式的所有根的绝对值之和，是图论的一个重要研究课题。刻画具有极值能量的图的结构是图能量研究的一个重要方面。目前研究极值能量的方法主要是拟序比较及图的运算。随着研究的进一步深入，出现了大量拟序不可比问题。这些拟序不可比问题的解决将会进一步完善图能量的理论研究，并为预测、合成新的化合物和新的药品方面提供理论基础。在前期工作的基础上，本项目利用组合迭代法并结合实分析的知识去考虑这类问题。我们首先解决了双圈图极小能量的一个公开问题，完全刻画了一类单圈图的极小能量图，同时考虑了Gutman等人提出的极大双圈图猜想，并有了部分进展。此外，本项目完全刻画了多部图的极值匹配能量和(n,m)-图的极小匹配能量。