混合图的Hermite谱研究

Spectral graph theory is an important research field of algebraic graph theory and combinatorial matrix theory,mainly studies the relationship between the spectral property and the structural property of graphs, discusses how the structural property of a graph is characterized by its spectral property.. 　The graph containing undirected edges and directed edges is called the mixed graph,which can be used to describe more general model.The theory of Hermitian spectra of mixed graphs is a new developing field on the spectral graph theory. In this project,we will try to study the following problems. (1)Exploring how much of a mixed graph can be reflect by its Hermitian spectrum; (2)Exploring how spectrum of a graph changes when its some edges are orientated; (3)Studying the differences between the Hermitian spectra of mixed graphs and the spectra of simple graphs; (4)Characterizing the structural property of a mixed graph, specially directed structural property, by its Hermitian spectrum. The innovation of the project embody in the following.We try to display the the essential relationship between the Hermitian spectral property and the structural property of mixed graph from the study of the difference between Hermitain spectrum of mixed graph and spectrum of simple graph, by recognizing the directly informations of edges by corressponding entries of Hermitian matrix.. This project should enrich the spectral graph theory, and promote the development of algebraic graph theory.

谱图理论是代数图论和组合矩阵论的一个重要研究领域,其主要目标是建立图的谱性质与图的结构性质之间的联系,用图的谱性质来刻画图的结构性质.. 混合图是既含有向边又含有无向边的图,可以描述更一般的现实模型.混合图的Hermite谱是近年来谱图理论新的研究方向.本项目的主要研究内容包括:(1)探讨混合图的Hermite谱反映混合图结构性质的程度;(2)探讨图的定向对Hermite谱参数的影响;(3)探讨混合图Hermite谱性质与简单图谱性质之间的差异;(4)混合图(有向)结构性质的Hermite谱刻画.本项目的特色在于:从Hermite谱性质与简单图谱性质之间差异的角度,通过Hermite矩阵对称位置上元素的共轭性质识别混合图中有向边的信息,揭示Hermite谱性质与有向结构性质之间的本质联系.. 本项目将拓广谱图理论的研究内容,推动代数图论的发展.

谱图理论是代数图论和组合矩阵论的一个重要研究领域,其主要目标是建立图的谱性质与图的结构性质之间的联系,用图的谱性质来刻画图的结构性质. . 本项目主要讨论了混合图与超图谱的一些相关问题，建立图的谱性质与其结构性质之间的若干联系。具体而言，在混合图的谱方面，明确提出ODHS图的概念用于谱刻画混合图定向的程度，刻画了所有的双圈ODHS图；研究了混合图在局部扰动下谱半径的变化情况，刻画了谱半径具有若干极端性质的混合图类；刻画了正惯性指数为1的混合图，在基础上确定了若干DHS图类；在Cayley混合图上刻画了一些谱确定的图。研究了超图的谱性质，从谱的角度刻画了超图的定向结构，将图与混合图谱的思想拓展至超图。