图的直径与斜秩研究

发布时间：2018-05-15 00:32

本文选题：定向图 + 斜秩　；参考：《安徽大学》2015年硕士论文

【摘要】：为了刻画图的结构性质,研究者引入多种图的矩阵,如邻接矩阵,Laplace矩阵,无符号Laplace矩阵等.这些矩阵都是实对称矩阵.近年来,定向图的斜邻接矩阵得到关注.它是一个反对称矩阵.给简单图G的每条边指定一个方向,得到一个定向图Gσ.根据边的方向,可以定义定向图Gσ的斜邻接矩阵.Cavers等人认为定向图的斜邻接矩阵的谱是区分同谱图的一种非常可能的方法,并提出了定向图的斜谱研究中的几个主要问题.建立定向图的结构性质与斜邻接矩阵的谱性质的联系,是谱图理论研究的最根本的问题.由斜谱产生的不变量,如斜秩,斜能量,斜谱半径得到广泛关注.定向图的斜秩定义为其斜邻接矩阵的秩.2009年BShader首次探讨了定向图的斜秩问题.2015年李学良和于桂海刻画了斜秩为2的定向图或斜秩为4的若干特殊图类.我们发现：图的直径至多为斜秩.因此,刻画直径等于斜秩的图显得尤为重要.本文刻画了直径与斜秩皆为4的定向图.本文的主要结构如下：在第一章中我们简单介绍了图谱和斜秩的发展以及本课题的现状,给出了基本概念和记号,以及本文的研究问题和主要结果.在第二章,我们介绍本文所需要引用的一些基本引理和推论,其次给出一个基本结论,即图的直径至多为斜秩,并探讨直径路的有关性质.第三章给出本文的主要结果,即刻画了直径与斜秩皆为4的定向图.为了获得该结果,我们证明了：直径路之外的点在直径路上至多有3个邻点.我们分别对这三种类型的点以及名种类型的点并存的情况开展讨论,最终获得本文的主要结果.
[Abstract]:In order to characterize the structural properties of graphs, researchers introduce a variety of matrices of graphs, such as adjacent matrices, unsigned Laplace matrices, and so on. These matrices are real symmetric matrices. In recent years, attention has been paid to the oblique adjacency matrix of directed graphs. It is an antisymmetric matrix. A direction is given to each edge of a simple graph G, and a directed graph G 蟽 is obtained. According to the direction of the edge, we can define the oblique adjacency matrix of the directed graph G 蟽. Cavers et al think that the spectrum of the skew adjacent matrix of the directed graph is a very possible method to distinguish the same spectral graph, and some main problems in the study of the skew spectrum of the directed graph are put forward. It is the fundamental problem to establish the relationship between the structural properties of directed graphs and the spectral properties of oblique adjacent matrices. The invariants produced by oblique spectrum, such as oblique rank, oblique energy and oblique spectral radius, have been paid more and more attention. The skew rank of a directed graph is defined as the rank of its skew adjoining matrix. In 2009, BShader first discussed the skew rank of the directed graph. In 2015, Li Xueliang and Yu Guihai characterized the directed graph with oblique rank 2 or some special graphs with skew rank 4. We find that the diameter of the graph is at most an oblique rank. Therefore, it is very important to depict graphs with diameter equal to oblique rank. In this paper, we characterize a directional graph with a diameter of 4 and an oblique rank of 4. The main structure of this paper is as follows: in the first chapter, we briefly introduce the development of atlas and skew rank and the present situation of this subject, and give the basic concepts and symbols, as well as the research problems and main results of this paper. In the second chapter, we introduce some basic lemmas and corollaries that need to be cited in this paper. Then we give a basic conclusion that the diameter of a graph is at most an oblique rank, and discuss the properties of the diameter path. In chapter 3, we give the main results of this paper, that is, we characterize the directed graph with diameter and oblique rank of 4. In order to obtain this result, we prove that there are at most three adjacent points in the diameter path. We discuss the coexistence of the three types of points and the famous types of points, and finally obtain the main results of this paper.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O157.5

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