矩阵空间的图同态与矩阵几何

A matrix graph is a simple graph that the vertex set is a set of matrices and an adjacency relation of vertices is defined by the rank of matrix. For example, bilinear forms graph,alternating forms graph,etc,in graph theory,they are important distance-regular graphs. Graph homomorphism is a core subject for research of algebraic graph theory,and the homomorphism of matrix graphs has important applications in graph theory and practical problems. However,it is difficult to characterize graph homomorphisms of a matrix graph,and this research has just started. Geometry of matrices is an algebra direction which is initiated by Loo-Keng Hua,and its basic problem is equivalent to researching graph isomorphisms of some matrix spaces. We find that the geometry of matrices has important application for characterizing graph homomorphisms of matrix graphs. Thus,this project makes cross and penetration research between algebraic graph theory and the geometry of matrices, and researches the following contents: A) graph homomorphisms(or chromatic number or independence number) for bilinear forms graphs,alternating forms graphs, Grassmann graphs and dual polar graphs over finite fields; B) strong graph homomorphisms for rectangular matrix graphs and Grassmann graphs over an infinite field (division ring) and the geometry of matrices; C) graph automorphisms for rectangular matrix graph and symmetric matrix graph over a Galios ring and their applications. This project has two innovative points: one is applying the theory of weighted semi-affine map to the research of graph homomorphisms, and the other is to solve several important problems for graph homomorphisms of matrix graphs. This project will make a contribution to the development and amalgamation of both algebraic graph theory and the geometry of matrices.

一个矩阵图是以某个矩阵空间作为顶点集,用秩定义顶点的邻接关系所得到的简单图,例如图论中的双线性型图、交错型图等重要的距离正则图.图同态是代数图论研究的核心课题之一,而矩阵图的同态在图论与解决实际问题中具有重要的应用,但它的刻画困难,其研究刚刚起步.矩阵几何是华罗庚开创的代数方向,其基本问题等价于研究各类矩阵空间的图同构.我们发现矩阵几何在矩阵图的同态刻画中有重要应用.因此本项目将代数图论与矩阵几何进行交叉和渗透研究,内容如下: A)有限域上双线性型图、交错型图、Grassmann图、对偶极图的同态、色数与独立数;B)无限域(体)上长方矩阵图、Grassmann图的强同态与矩阵几何;C)伽罗瓦环上长方、对称矩阵图的自同构及应用.本项目有两个创新点,一是应用"加权半仿射映射"理论研究图同态,二是解决几个重要的矩阵图的同态刻画问题.本项目将为代数图论与矩阵几何两个方向的发展、融合做出贡献.

图同态是代数图论研究的核心课题之一,矩阵几何是数学大师华罗庚开创的一个数学方向.本项目将代数图论与矩阵几何进行交叉,研究下列内容:有限域上矩阵图的同态;体上矩阵空间的图同态与矩阵几何;伽罗瓦环上矩阵图及其应用.本项目的一个创新点是刻画几个重要的图类的自同态. . 本项目取得重要的研究成果.回忆一个图是核如果这个图的自同态都是自同构,一个图是伪核如果这个图的每个自同态是一个自同构或者顶点着色.我们证明了一些重要的图是伪核,并给出了这些图是核的条件,从而完全刻画了它们的图自同态.例如:有限域上双线性型图、交错型图、格拉斯曼图、挠格拉斯曼图、5类对偶极图,等等. 我们还得到这些图的色数,团数,独立数的一些计数公式.应用加权半仿射映射,我们刻画了在较弱条件下的两个体上的两个长方矩阵空间之间的图同态.我们建立了剩余类环上双线性型图, 广义双线性型图和格拉斯曼图的理论,等等. 这些成果有重要的科学意义和应用价值,促进了代数图论与矩阵几何两个方向的发展.本项目发表论文16篇, 录用论文1篇, 其成果得到国内外学者的好评和多次引用.