关于图的几种指数的研究

发布时间：2018-05-13 16:29

本文选题：矩阵 + 代数连通度　；参考：《上海交通大学》2015年博士论文

【摘要】：关于图的指数和图的结构之间的关系的研究是图论的核心问题之一.本文主要研究的是图谱以及化学图论中一些指数和图的结构之间的联系,包括代数连通度、Zagreb指数以及Estrada指数.这些指数在化学图论的研究中具有重要的意义并被广泛应用.在代数连通度方面,刻画了给定顶点数和最大度的树的代数连通度最小的所有图.同时求出了给定顶点数和最大度的树的代数连通度的下界.在Zagreb指数方面,研究了给定度序列的图的第二Zagreb指数的性质和图的结构,并刻画了一些令第二Zagreb指数达到最大值的图的结构.同时,这些结论推广了刘木伙和柳柏濂的关于给定度序列的树和单圈图的结论.另外,我们还详细研究了给定度序列的双圈图的第二Zagreb指数.在Estrada指数方面,研究了给定顶点数和直径的树中Estrada指数最大的图的性质,并且刻画了给定顶点数和直径的树中具有最大Estrada指数的所有图.
[Abstract]:The study of the relationship between graph exponent and graph structure is one of the core problems in graph theory. In this paper we mainly study the relationship between some indices and the structure of graphs in graph theory including algebraic connectivity Zagreb index and Estrada index. These indices are of great significance in the study of chemical graph theory and are widely used. In terms of algebraic connectivity, we characterize all graphs with minimal algebraic connectivity of a tree with a given number of vertices and maximum degree. At the same time, the lower bound of algebraic connectivity of a tree with given number of vertices and maximum degree is obtained. In the aspect of Zagreb exponent, the properties of the second Zagreb exponent and the structure of the graph of a given degree sequence are studied, and the structures of some graphs with the maximum value of the second Zagreb exponent are characterized. At the same time, these results generalize the results of Liu Mu-huo and Liu Po-lien on trees and unicyclic graphs of given degree sequences. In addition, we study in detail the second Zagreb exponent of the bicycle graph of a given degree sequence. In terms of Estrada exponents, we study the properties of graphs with the largest Estrada exponent in a tree with a given number of vertices and diameters, and characterize all graphs with the largest Estrada exponent in a tree with a given number of vertices and diameters.
【学位授予单位】：上海交通大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O157.5

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