关于一致超图的谱半径

发布时间：2018-06-19 05:23

本文选题：张量 + 谱半径　；参考：《安徽大学》2015年硕士论文

【摘要】：图是研究离散对象二元关系系统的结构性质的一个重要的数学模型.作为图的推广,超图是有限集合的子集系统,更加能够反映现实世界的对象之间的复杂多元关系.金芳蓉,冯克勤,李文卿,Friedman, Wigderson, Rodriguez,陆临渊和彭兴等引入超图的邻接矩阵或Laplace矩阵研究超图的若干性质.但是,超图的每条边是由两个以上的点所确定,所以上述定义没有直接地反映超图的结构性质.Lek-Heng Lim和祁力群,张恭庆等独立地引入了张量的特征值和特征向量及相关性质.至此,一致超图的张量表示(邻接张量,Laplace张量和无符号Laplace张量)的特征值和特征向量得到广泛研究.2014年,李红海,祁力群和邵嘉裕讨论了树的幂图的最大和最小谱半径.本文将探讨更一般的问题,即给定边数的一致超图的最大谱半径.我们证明了：在给定点数与边数的所有连通的k-一致超图中,谱半径达到最大的图一定包含一个与其它所有点都邻接的点.由此,我们分别刻画具有最大谱半径的单圈超图和给定围长的线性或幂单圈(或双圈)超图.本文的主要结构如下：在第一章中我们简单介绍了图和超图谱图理论的发展过程,给出了基本概念和记号,以及本文的研究问题和主要结果。第二章首先介绍非负张量的Perron-Frobenius定理,然后刻画了超图在其分支迁移后的谱半径变化情况。第三章刻画了具有最大谱半径的单圈超图和给定围长的线性或幂单圈(或双圈)超图.
[Abstract]:Graph is an important mathematical model for studying the structural properties of discrete object binary relation system. As a generalization of graphs, hypergraphs are subsets of finite sets, which can reflect the complex multivariate relationships between objects in the real world. Jin Fangrong, Feng Keqin, Li Wenqing Friedman, Wigderson, Rodriguez, Lu Linyuan and Pengxing introduce adjacent matrices or Laplace matrices to study some properties of hypergraphs. However, each edge of a hypergraph is determined by more than two points, so the above definition does not directly reflect the structural properties of hypergraphs. Lek-Heng Lim and Qili groups. Zhang Gongqing and others independently introduce the eigenvalues, eigenvectors and related properties of Zhang Liang. So far, the eigenvalues and Eigenvectors of the Zhang Liang representations of uniform hypergraphs (adjacent to Zhang Liang Laplace Zhang Liang and unsigned Laplace Zhang Liang) have been extensively studied. In 2014, Li Honghai, Qili Group and Shao Jiayu discussed the maximum and minimum spectral radii of power graphs of trees. In this paper, we will discuss a more general problem, that is, give the maximum spectral radius of a uniform hypergraph with a fixed number of edges. We prove that in all connected k-uniform hypergraphs with a given number of points and edges, a graph with a maximum spectral radius must contain a point adjacent to all other points. Therefore, we characterize the single cycle hypergraphs with the maximum spectral radius and the linear or power unicyclic (or double cycle) hypergraphs with given girth respectively. The main structure of this paper is as follows: in the first chapter, we briefly introduce the development process of graph and hypergraph theory, and give the basic concepts and symbols, as well as the research problems and main results of this paper. In chapter 2, we first introduce the Perron-Frobenius theorem of nonnegative Zhang Liang, and then characterize the variation of spectral radius of hypergraphs after branch migration. In chapter 3, we describe the monocyclic hypergraphs with the maximum spectral radius and the linear or power unicyclic (or double-cycle) hypergraphs with given girth.
【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

【共引文献】