混合图的Hermite矩阵表示及其谱研究

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 early work concerning the spectrum of mixed graph mainly focus on symmetrical matrix of mixed graph and its spectrum. However, the directed structural information of mixed graph cannot be represented by its current matrix, so it is very difficult to characterize its directed structural properties by spectrum. This project would attempt to establish the matrix(Hermitian matrix) which can reflect the directed structure of mixed graph well, and study the relationship between the directed structural property of mixed graph and its spectral property by algebraic and combinatorial approaches, mainly include: (1) establishing the effective Hermite matrix of mixed graph to reflect its directed structure; (2) characterizing the relationship between the directed structural property of mixed graph（strong connectivity, unilateral strong connectivity， directed path, directed cycle） with its spectral property. This project should enrich the spectral graph theory, and promote the development of algebraic graph theory and combinatorial matrix 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。关于混合图的谱研究，前期工作主要集中在其对称矩阵表示及其谱研究。这种矩阵表示没有体现边的方向信息，从而很难用谱来描述混合图的有向结构性质。本项目旨在建立一种能够反映混合图有向结构信息的矩阵表示，即Hermite 矩阵表示，利用代数和组合分析的方法，建立这种矩阵表示的谱性质与混合图有向结构性质之间的联系。主要内容包括：（1）混合图有效的Hermite矩阵表示；（2）混合图的谱性质与有向结构性质（如强连通性、单向强连通性、有向路和有向圈等）的联系。. 本项目将拓广谱图理论的研究内容，推动代数图论和组合矩阵论的发展。

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。. 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。本项目建立了混合图更为一般的, 有效的Laplace矩阵表示, 并给出这种矩阵表示的组合刻画, 从组合的角度理解并计算了这种矩阵的行列式. 建立混合图强连通性与混合图H-邻接矩阵之间的联系, 从矩阵角度给出了混合图强连通的等价描述.