代数图论在分子拓扑指数中的应用

It is very important for the molecular topological index in the applications of chemistry, biology, physics and so on. This project study molecular topological index by the methods of algebraic graph theory, and further improve and broaden the results of Estrada index. The task of this project will focus on the following five aspects: .1) the extremal values or the ordering of some clasess of graphs by their Estrada indices; .2) the better upper and lower bounds; .3) the perturbations of Estrada indices throught the operations on the graphs;.4) the relation between the Estrada index and some graph''s parameters such that they have the same or inverse variation; .5) the relation between the original graph''s Estrada indices and the new graph''s Estrada index.

分子拓扑指数在化学、生物和物理等领域有着重要的应用。本项目运用代数图论的方法研究其中一类分子拓扑指数：Estrada指数，并对其研究结果进一步拓广和完善。研究内容主要集中在如下五个方面：.1) 研究一些分子图类的Estrada指数的极值问题并刻画其相应的极图，进一步对Estrada指数进行排序；2) 获得Estrada指数更优的界；.3) 研究Estrada指数的扰动问题；4) 建立Estrada指数与图的其它参量之间的联系；.5) 研究原图的Estrada指数与新图的Estrada指数与之间的联系。

本项目讨论了二类问题：1）探讨了距离矩阵谱半径的极值问题，利用距离谱半径的界得到了连通图包含哈密尔顿路及其哈密尔顿圈的条件；2）把矩阵的因式分解理论应用到了自然数集及正整数集上矩阵半群中,给出了这类矩阵半群的一些重要性质和特征,并对文献文献[Baeth N. et al., Number Theory of Matrix Semigroups,[J], Linear Algebra Appl. 2011, 434 (3): 694-711.]中所提出的公开问题1-4做出了部分解答.