The Relation Between Rank And Girth Of Simple Graph And The Relation Between Skew-rank And Girth Of Directional Graph

In recent years,algebraic graph theory as an important branch of mathematics ha-s set off a wave of academic research upsurge,graph as the basic element of graph theo-ry research,so that the rank of simple graph,oblique adjacency matrix of directional gr-aph,circle graph girth has won the favor of many scholars,and many excellent researc-h results have been obtained.The rank of a cycle-containing graph G refers to the rank of its adjacency matrix A(G),the girth is the length of the shortest circle in the graph G,and S(G~σ)is the obliq-ue adjacency matrix of the directed graph G~σ.The skew-rank of the directed graph G~σi-s defined as the rank of s(G~σ),and,r(G)、g(G)、sr(G~σ)respectively.In this paper,t-he rank of simple graph,the skew-rank of directional graph and the girth of circle grap-h are related to the excellent research theorem results.Through sorting and analysis,th-e relationship between the rank and girth of simple graph is r(G)≥g(G)-2 the relatio-n between the skew-rank and girth of directional graph is sr(G~σ)≥g(G)-2 and the nece-ssary and sufficient conditions of r(G)=g(G)-2、sr(G~σ)=g(G~σ)-2 are demonstrate-d.A nd studied the relationship between rank and girth is r(G)≥g(G),skew-rank and girth is sr(G~σ)≥g(G~σ)when the pendent point is included.This article contains four chapters,as follows:Chapter 1:The introduction of resear-ch background,research status and basi-c concepts.Chapter 2:The basic lemma,the relationship between rank and perimeter lengt-h of simple graphs,the proof of sufficient necessity for equality,the relationship betwe-en rank and girth of simple graphs including pendent points,and the graphs included i-n equality.Chapter 3:The excellent theorem results of directional graphs,the relationship b-etween skew rank and girth of directional graphs and the proof of sufficient necessity-y when they are equal,the relationship between skew rank and girth when they include-e pendent points,and which graphs are included when they are equal.Chapter 4:the conclusion part,summarizes the main content of the paper,and pu-ts forward the future learning objectives and research innovation direction in this arti-cleFigure:nine Reference:eighty-one...