Research On The Rank Of Simple Graphs With Cut Points

As a branch of mathematics,graph theory has been widely developed in various fields.In order to study the properties of graphs,the concept of adjacency matrix is introduced.The rank of a graph is defined as the rank of its adjacency matrix,which is the number of nonzero eigenvalues in the spectrum.The rank of graphs has been applied in many fields.In fact,the research on the rank of graphs has become a hotspot in algebraic graph theory.A simple graph is one that contains neither parallel edges nor self-loops.In this paper,we study the rank of simple graphs with cut points,characterize the connected reduced graphs with cut points of rank 6,and give the construction method of connected reduced graphs with pendent points of rank 6,moreover,the connected reduced tricyclic graphs with rank 6 and rank 5 with cut points or pendent points are completely characterized by using these results,the concrete graph class is given.The structure of this paper is divided into the following chapters:The first chapter introduces the research background and significance of graph with cut points,to understand the research trends of domestic and foreign scholars,then some basic concepts of graphs and the research problems and main conclusions of this paper are given.In Chapter 2,we study the rank of simple graphs with cut points.Firstly,by using the cut points lemma,we discuss some cases of the rank of branches without cut points,and characterize the connected reduced graphs with cut points of all rank 6,then by using the construction method of simple graphs with pendent points,the characterization of the connected reduced graphs with pendent points of rank 6 is given.In the third chapter,the 2-connected reduced graph of rank 6 without cut and pendent points is determined by ear decomposition theorem.Finally,we summarize the previous discussion and give the applied characterizations of the connected reduced tricyclic graphs of rank 6.In Chapter 4,on the basis of the related conclusions,we also use the cut points lemma to characterize the reduced tricyclic simple signed graphs with cut points or pendent points of rank 5.The fifth chapter is the conclusion,summarizes the article and explains the innovation and the insufficiency.Figure[13]Table[0]Reference[69]...