图的距离谱和距离拉普拉斯谱的研究

发布时间：2018-01-05 09:39

本文关键词：图的距离谱和距离拉普拉斯谱的研究　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：图论是一门非常重要的科学：它广泛应用于各个领域,如计算机网络.生命科学.生物化学.组合优化.分子理论等.而图谱理论又是图论中的一个非常重要的分支.多年来对图谱理论的研究一直处于非常活跃的状态.也取得了许多非常成熟和重要的成果及应用.本文主要应用图论和代数相结合的方法以及矩阵的有关理论性质来研究两种具体的谱：距离谱和距离拉普拉斯谱.在前人研究的基础上得到一些比较有意义的结果.顺便解决了前人提出的一些猜想.本文共分为三个章节.第一章是绪论部分.第二章主要做了关于距离谱的一些成果.第三章主要做了关于距离拉普拉斯谱的一些成果.下面我们分别简要介绍一下这三章的主要内容.(一)在第一章.第一小节中,我们简要回顾了图论的起源,图论的发展过程,然后介绍了图谱理论研究经常用到的一些方法和技巧.在第二小节中.我们介绍了本文用到的一些基本概念和记号.还有一些特殊记号,此处没有介绍到的我们会在有关章节做出具体介绍.在第三小节中,我们简要介绍了本论文所涉及到的问题和问题的进展情况.(二)在第二章中的第一小节,我们给出了对角元素全为零的非负不可约矩阵谱半径的两个紧的上界.并分别刻画了达到上界的充要条件.作为推论.我们给出了距离矩阵谱半径的两个紧上界.并分别刻画了达到上界的极图.在第二小节中.我们证明了当D是n个顶点的图G的距离矩阵时.对任意给定的非负整数k,当n充分大时,有D的第n-k大特征值λn-k(D)≤1成立.从而回答了文献[41]提出的问题.在第三小节中,我们刻画了距离矩阵特征值-1的重数分别为n-i(i=1,3,4)时的极图.在第四小节中,我们刻画了完全分裂图是距离整谱图的充要条件.(三)在第三章中的第一小节,我们给出了距离拉普拉斯谱半径的下界以及第二小的距离拉普拉斯特征值的上界.并给出了在某些图类上的应用,刻画了相应的极图.在第二小节中,我们给出了某些图类距离拉普拉斯谱展的下界.并刻画了相应的极图.在第三小节中,我们给出了距离拉普拉斯谱半径重数的上界.并刻画了相应的极图,从而证实了Aouchiche和Hanson在文献[3]中提出的一个猜想.
[Abstract]:Graph theory is a very important science: it is widely used in various fields, such as computer networks. Life sciences. Biochemistry. Combinatorial optimization. Molecular theory. The graph theory is a very important branch of graph theory. Over the years the study of graph theory has been a very active state. Have a lot of very mature and important achievements and application of relevant theory of nature. This paper combines the application of graph theory and algebra and matrix of two specific spectrum: the distance spectrum and Laplasse spectrum. The distance based on the previous studies, some of the more meaningful results. The way to solve some of the previous conjecture this paper is divided into three chapters. The first chapter is the introduction part. The second chapter made some achievements on the distance spectrum. The third chapter is mainly done on the Laplasse spectrum of some distance We have the following results. Briefly introduce the main content of the three chapter. (a) in the first chapter. In the first section, we briefly review the development process of graph theory, graph theory, and then introduces some methods and techniques of theoretical research is often used. In the second section, we introduce some basic. The concept and mark used in this paper. There are some special mark here, we will not introduce to make specific introduction in relevant chapters. In the third section, we briefly introduced the progress and problems involved in this study. (two) in the first section of the second chapter, we give the diagonal elements all two tight upper bound of non negative zero irreducible matrix spectral radius. And we characterize the necessary and sufficient conditions to the upper bound. As a corollary. We give two a tight upper bound on the spectral radius of the matrix. The distance and moment respectively Draw graphs achieving the upper bound. In the second section. We prove that when D is the distance matrix of a graph on n vertices of G. Non negative integer k for any given, when n is sufficiently large, the n-k characteristics of D value n-k (D) = 1 was established. In order to answer the question raised by [41]. In the third section, we characterize the distance matrix eigenvalue multiplicity -1 n-i respectively (i=1,3,4). When the pole figure in the fourth section, we characterize the complete split graph is the distance from the necessary and sufficient conditions of integral graphs. (three) in the first section of the third chapter. Laplasse, we give lower bounds on the spectral radius of the distance and the second smallest eigenvalue of the Laplasse distance bound. And gives the application in some graphs, depicts the corresponding extreme graphs. In the second section, we give a lower bound of some graphs from Laplasse spectrum show. And characterize the corresponding extreme graphs in the third. In the section, we give the upper bound of the multiplicity of the distance from the Laplasse spectral radius, and characterize the corresponding polar graph, thus confirming a conjecture proposed by Aouchiche and Hanson in document [3].

【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O157.5

【相似文献】