Research Of Rank On Several Kinds Of Directional Graphs

The application of graph theory is very wide and plays an important role in solving problems in physics,chemistry and computer networks.Graph theory is also one of the important topics in algebraic graph theory.People introduce a variety of matrices,such as Laplacian matrix,adjacency matrix,incidence matrix and so on,by studying the properties of the spectrum of graphs,and then in the process of studying the properties and structure of graphs,these matrices are closely related to graphs.At present,the most studied and most fruitful matrix is adjacency matrix.In this paper,according to the directed graph obtained by simple graph,G is set as an undirected graph,and each edge of G is given a direction,that is,the original undirected edge u v is replaced by arc u→v or v?u,and the obtained graph is called the directed graph,denoted as D（G）.Let’s call it D.Directed complete graphs are a class of simple graphs.A simple graph is a graph without heavy edges and cycles.An undirected simple graph is denoted as G=（V,E）,and the vertex set V is divided into two non-empty subsets,and the two non-empty subsets are denoted as X and Y.The two vertices _ix and y_j associated with each edge in the graph belong to the two disjoint vertex sets respectively,and the graph G is called bipartite graph.Let any point in the set of points X be adjacent to any point in the set of points Y,but any two points in X are not adjacent and any two points in Y are not adjacent,then G is said to be a complete bipartite graph.In the same way,the complete three-part graph definition is obtained.This paper mainly uses the property of twin points,that is,if there is no twin point pair in an oriented graph D,the graph is called reduced graph,denoted as T（D）.Deleting a point in a twin point pair will not affect the change of graph rank（from the perspective of matrix,the row transformation of matrix does not change the rank of matrix）.You transform a complex graph into a simple graph.In this paper,the rank r（D）of graph D is defined as the rank of the adjacency matrix mainly according to the adjacency matrix.Oriented graph based on complete bipartite graph,that is,every edge x_i y_j is assigned an orientation:x_i→y_j orx_i?y_j;Oriented graph complete tripartite graph,that is,every edgex_i y_j,x_i z_k andy_j→z_k is assigned an orientation:x_i→y_j orx_i?y_j,x_i→z_k orx_i?z_k,y_j→z_k ory_j?z_k.By deriving the theorems and properties of subgraphs and twin points,the ranks of several classes of directed complete bipartite and tripartite graphs are characterized.This thesis is divided into four chapters.The first chapter mainly introduces the related research background and basic concepts.The second chapter introduces some relevant lemmas and research progress,and describes the situation of oriented complete bipartite graphK_2,n and the rank ofK_3,n,the third chapter describes the situation of oriented complete tripartite graphK_1,1,n,and the fourth chapter is a summary of the full text,and points out the innovation and deficiency of this paper.Figure:fifty-six Reference:fifteen...