【作者】 胡丹；

【导师】 李学良；

【作者基本信息】 西北工业大学 ， 应用数学， 2019， 博士

【摘要】 随机图论由Erd?s和Rényi在上个世纪50年代创立,它主要运用概率论和随机过程的理论与方法来研究图的结构性质和代数性质.随机图谱理论是随机图论的一个重要组成部分,主要研究随机图对应的随机矩阵的特征值和特征向量的极限性质,它不仅有着重要的理论价值而且在物理及化学等诸多领域都有着广泛的应用.本文基于随机多部图、随机混合图和随机定向图三类随机图模型,研究其对应的若干随机矩阵的谱性质.主要研究成果如下:1.研究了随机多部图的谱以及图的一些不变量与图谱之间的关系,通过分析随机对称矩阵谱的极限,给出了随机多部图的Laplacian谱估计,得到了随机多部图的Laplacian能量、Laplacian Estrada指标以及冯·诺依曼熵的渐近上下界.2.研究了随机混合图的Hermitian邻接矩阵和规范化Hermitian Laplacian矩阵的谱性质,首次提出了随机混合图模型并给出了与其对应的Hermitian邻接矩阵和规范化Hermitian Laplacian矩阵的极限谱分布—半圆律;通过Hermitian邻接矩阵的极限谱分布得到了随机混合图Hermitian能量的渐近表达式;给出了 Hermitian邻接矩阵最大特征值的渐近表达式以及其它特征值的渐近界估计,进一步得到了随机混合图Hermitian谱矩的渐近表达式.3.研究了广义随机混合图的Hermitian邻接矩阵和规范化Hermitian Laplacian矩阵的谱,改进了随机矩阵中的Bernstein不等式,给出了一个关于相互独立的随机Hermitian矩阵之和的概率不等式,通过该概率不等式给出广义随机混合图的Hermitian邻接矩阵谱半径的渐近界估计,并给出了规范化Hermitian Laplacian矩阵特征值的近似估计.4.研究了随机定向图的斜邻接矩阵和斜Randic矩阵的谱半径,将随机定向图的斜邻接矩阵和斜Randic矩阵写成若干个随机矩阵之和,运用一个关于相互独立的随机矩阵之和的概率不等式来给出这两个矩阵谱半径的渐近界估计.上述研究成果揭示了随机多部图、随机混合图以及随机定向图三类随机图的谱性质,既丰富了随机图谱的理论研究,又拓展了随机矩阵理论中的概率不等式和Wigner半圆律等重要结果.更多还原

【Abstract】 Random graph theory was founded by Erdos and Renyi in 1950s.It mainly studies the relationships between the spectral property and the structural property of graphs with the tools of probability theory and stochastic process theory.Spectral random graph theory is an important part of random graph theory.It mainly focuses on the limiting behavior of the eigenvalues and eigenvectors of various random matrices associated with random graphs.It not only has important theoretical value but also has extensive applications in many fields such as physics and chemistry.In this dissertation,based on the random multipartite graph model,the random mixed graph model and the random oriented graph model,the spectral properties of several corresponding random matrices are studied.The main research results are as follows:1.The spectra of random multipartite graphs and the relationship between some invariants and the spectra of graphs are studied.By analyzing the the limiting behavior of the spectra of random symmetric matrices,we estimate the eigenvalues of the Laplacian matrix of random multipartite graphs,and establish asymptotic lower and upper bounds for the Laplacian energy,the Laplacian Estrada index and the von Neumann entropy of random multipartite graphs respectively.2.The spectral properties of Hermitian adjacency matrices and normalized Hermitian Laplacian matrices of random mixed graphs are studied.The concept of the random mixed graph model is proposed for the first time.We prove that the limiting spectral distributions of Hermitian adjacency matrices and normalized Hermitian Laplacian matrices of random mixed graphs are Wigner’s semicircle law.The asymptotic expression of the Hermitian energy of random mixed graphs is given by the limiting spectral distribution of Hermitian adjacency matrices.Moreover,the asymptotic expression of the largest eigenvalue of the Hermitian adjacency matrix and the asymptotic bounds of other eigenvalues are obtained,and then the asymptotic estimation of the Laplacian spectral moment of random mixed graphs is given.3.The spectra of Hermitian adjacency matrices and normalized Hermitian Laplacian matrices of general random mixed graphs are studied.We establish a stronger version of Bernstein’s inequality on random matrices,and present a new probability inequality for sums of independent random Hermitian matrices.Using this probability inequality,we estimate the asymptotic bound of the spectral radius of the Hermitian adjacency matrix,and give an approximate estimation of the eigenvalues of the normalized Hermitian Laplacian matrix.4.The spectral radii of skew adjacency matrices and skew Randic matrices of random oriented graphs are studied.We express the skew adjacency matrix and the skew Randic matrix by sums of independent random matrices,and then establish asymptotic upper bounds for the spectral radii of skew adjacency matrices and skew Randic matrices of random oriented graphs by a probability inequality for sums of independent random matrices.The above research results reveal the spectral properties of random multipartite graphs,random mixed graphs and random oriented graphs.It not only enriches the theoretical research of random graphs,but also extends the important results such as probability inequalities and Wigner’s semicircle law in random matrix theory.更多还原