图谱理论中若干问题的研究

The graph spectral theory is an important research subject in combinatorics and algebraic graph theory, this project will study some problems on the theory and its application of spectra of graphs. And the main task of this project will focus on the followings: in the aspect of spectra of graphs, we try to search a general and convenient approach to the order of the spectral radii of graphs, which will transfer the comparison of spectral radii to the comparision of some parameters of graphs, we also investigate the method to construct new graphs determined by their spectra (respectively, new integral spectral graphs) from the original graphs determined by their spectra (respectively, original integral spectral graphs), and study the extremal graphs with the maximum (minimum) spectral radii in the class of connected graphs with given degree sequences，based on which we shall try to discover new majorization theorems in the spectra of graphs; in the aspect of the applications of spectra of graphs, we focus on some problems of different kinds of chemical indices, which are closely connected with the spectra of graphs. In this direction, we shall investigate the general and convenient approach to the order of some chemical indices, study the extremal graphs with the maximum (minimum) chemical indices in the class of connected graphs with given degree sequences，and we shall also try to discover new chemical indices which obey the majorization theory.

图的谱理论是组合数学与代数图论中研究的一个重要课题，本课题将对图谱的理论及其应用中的若干问题进行研究，研究的主要问题如下：在图谱理论方面，我们将致力于寻找一般且简便的谱半径的排序方法，尝试通过图本身的某些参数的比较来达到谱半径的比较目的，探索由已知"由谱所确定"的图类(整谱图)出发如何构造新的"由谱所确定"的图类(整谱图)的方法，尝试研究给定度序列的图类中具有最大(最小)谱半径的极图刻划，并以此为基础来发掘新的图谱的优超定理；在图谱应用方向，将重点研究与图谱有密切联系的各类化学指数的若干问题，如探索某些化学指数的一般且简便的排序方法，尝试刻划给定度序列的图类中具有最大(最小)化学指数的极图，并寻找服从优超理论的新的化学指数.

本项目着重于图谱的极值理论的研究，研究的内容集中于研究图谱的优超理论和谱半径的标尺定理，由谱所确定的图类，以及与图谱密切相关的化学指数的极值结论与方法。课题所研究的内容属于信息科学与数学的交叉课题，在计算机科学、经济分析和化学结构中有广泛的应用。经过课题组成员的共同努力，我们较好地完成了研究计划中预定的各项研究任务，在一些较为重要的课题的研究中取得了较大的进展。在该课题资助下，课题组成员共发表论文19篇，其中18篇被SCI收录(包括2篇研究综述)。在课题所取得的成果的基础上，课题主持人完成了研究专著《图谱的极值结论》，该专著受到广东省优秀科技专著出版基金会立项资助，计划在2016年出版。