Some Research Of Rank On Mixed Graph

Graph theory is an important research field in algebraic graph theory and combi-natorial matrix theory.It develops rapidly in recent decades,and attract many attentions and favors of many researchers.All kinds of matrices corresponding to graphs are im-portant research directions and contents in graph theory.In addition to studying the spectrum of graphs,the rank and nullity of graphs are also concernedIn 2007,Cheng et al.proposed to characterize graphs with given rank first,and then they depicted undirected graphs with rank 4,5.Further,researchers began to study directed graphs and mixed graphs.There are many kinds of adjacency matrix of direct-ed graphs and mixed graphs.In this paper,we discuss two kinds of common adjacency matrix:adjacency matrix A of directed graphs and Hermitian adjacency matrix H of mixed graphIn this paper,we mainly characterize the oriented graphs of rank 2 and the mixed graphs with cut points of rank 4.The conclusions are as follows:Theorem 2.1:A connected oriented graph D has rank 2 if and only if D,or DT,is obtained from one of the oriented graphs in fig2-1 by vertex-multiplicationTheorem 3.15:Let T is a reduced mixed graph tree and r(T)=4,then the underlying graph of T is P4,p,q or P5,p,q,1≤p≤3,1≤q≤3Theorem 3.16:Let G is a connected reduced mixed graph,G is not a tree,u is a pendant graph of G,v is the neighbor of u.If r(G)=4,then G-u-v is switching equivalence with Kα,b or Ca,b,c.Theorem 3.17:Let G is a connected reduced mixed graph with a cut vertex v,G is not a tree and it don’t have pendant vertex.r(G)=4 if and only if G-v only have two components G1,G2,and either G1+v and G2+v switching equivalence with a graph in fig3-2,or G is switching equivalence with Γ1 or Γ2(see fig3-3).There are five figures in this paper and 53 References.