混合图Hermite同谱类的个数

发布时间：2018-05-05 06:09

本文选题：混合图 + Hermite邻接矩阵　；参考：《安徽大学》2017年硕士论文

【摘要】：谱图理论是代数图论和组合矩阵论共同关注的一个重要研究方向.混合图的Hermite谱理论是近年来谱图理论一个热点研究课题,主要通过建立混合图Hermite谱性质与混合图结构性质之间的联系,用Hermite谱性质刻画混合图的结构性质.为实现这一目标,首先要考虑的问题是混合图的Hermite谱能在多大程度上反映混合图,即混合图的Hermite谱确定问题.对于不能由Hermite谱确定的图类,确定或界定同谱类的个数也是一个自然且重要的问题.本文即讨论了两类给定底图的混合图的同谱类个数问题,确定了双圈混合图所有可能的同谱类个数,并依据同谱类的个数对双圈图进行分类,给出了平面κκ圈混合图同谱类个数一个可达的上界.本文的组织结构如下.第一章首先介绍了图论和谱图理论的研究背景,其次介绍了常用的概念和符号,最后介绍了文章所研究的问题、目前进展及本文所得的主要结果.第二章首先介绍了本节所需的预备知识,然后确定了给定底图的双圈混合图的同谱类个数,并依据同谱类的个数对双圈图进行分类.第三章首先介绍了本节所需的预备知识,然后给出了给定底图的平面κ-圈混合图同谱类个数的上界,并构造一类可以达到该上界的平面κ-圈混合图.
[Abstract]:Spectral graph theory is an important research direction of algebraic graph theory and combinatorial matrix theory. The Hermite spectrum theory of mixed graphs is a hot topic in spectral theory in recent years. By establishing the relationship between the Hermite spectral properties of mixed graphs and the structural properties of mixed graphs, the structural properties of mixed graphs are characterized by Hermite spectral properties. In order to achieve this goal, the first question to be considered is the extent to which the Hermite spectrum of the mixed graph can reflect the mixed graph, that is, the Hermite spectrum determination of the mixed graph. For graph classes which cannot be determined by Hermite spectrum, it is also a natural and important problem to determine or define the number of isospectral classes. In this paper, we discuss the number of isospectral classes of mixed graphs of two given base graphs, determine the number of all possible isospectral classes of bicyclic mixed graphs, and classify bicyclic graphs according to the number of isospectral classes. A reachable upper bound for the number of isospectral classes of a mixed 魏 -cycle graph is given. The organizational structure of this paper is as follows. In the first chapter, the research background of graph theory and spectral graph theory is introduced, then the commonly used concepts and symbols are introduced. Finally, the problems studied in this paper, the current progress and the main results obtained in this paper are introduced. In the second chapter, we first introduce the necessary preparatory knowledge in this section, then determine the number of the same spectral classes of the double cycle mixed graphs of a given base graph, and classify the double cycle graphs according to the number of the same spectral classes. In chapter 3, we first introduce the necessary preparatory knowledge in this section, then we give the upper bound of the number of the same spectral classes of the plane 魏 -cycle mixed graphs of a given base graph, and construct a class of planar 魏 -cycle mixed graphs which can reach the upper bound.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5

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