Research On The Relation Between The Skew-Rank,Girth And Diameter Of An Oriented Graph

The rank and structure of graphs are important research contents in graph t heory.At the moment,the research on the algebraic properties of graphs is mai nly to analyze the adjacency matrix,Laplace matrix and other real symmetric m atrices of graphs,and then study the structure of graphs and the relationship be tween them.Now,the research on the relation between rank,girth and diameter of simple graphs has achieved some excellent results,and the research on the relation between skew-rank,girth and diameter of oriented graphs has also attra cted scholars’ attention.The skew-rank of graph Gσ is defined to be the rank of its skew-adjacency matrix,the girth of the graph is defined to be the length of its shortest cycle of underlying graph G,and the diameter is defined to be the longest distance between two vertices of its underlying graph.They are denoted by sr(Gσ),g(G),d(G),respectively.By studying the anti-symmetric of skew-adjacency matrices,we can ensure that their skew-rank are even.Therefore,exploring the interaction between skew-adjacency matrices and structural parameters has become an important task for us to explore oriented graphs.It is well known that the skew-adjacency matrix is skew-symmetric,so the skew-rank obtained must be an even.The study of the relations between skew-rank and several structural parameters is to study the relations between skew-adjacency matrix and several structural parameters.In this paper,all oriented graphs of sr(Gσ)=g-(G)-1 and sr(Gσ)=g(G)are described by the operation of skew-adjacency matrix and the relation between the obtained sub-graph and the original graph.It is known that the diameter does not exceed the skew-rank from the relation between the diameter and the skew-rank of the oriented graph,so there is a most one path Pn+1 given the skew-rank of the graph n.We add vertices on the underlying graph of the diameter path and discuss them according to different vertices,then characterize the oriented graphs with skew-rank n which attain the maximum diameter n.We also bound the skew-rank of oriented complete graphs with given order.The major researches of this article are as follows:In this article,we firstly introduce the research background and significance of knowledge associated with diagram theory,the current research state of the topic components,and also give the appropriate elementary concepts and symbols.Secondly,this article delves into some basic laws,inferences,and related evidence,and applies them to various forms of directed graphs with skew-rank equaling to girth to obtain results.Thirdly,according to the research on diameter path,we characterize the oriented graphs with skew-rank n which attain the maximum diameter n.We also bound the skew-rank of oriented complete graphs with given order.Finally,we summarize the research of this article and state the shortcomings as well as possible directions for future research.Figure:nineteen Reference:eighty-one...