On The Relation Between The H-rank Of A Mixed Graph And The Matching Number Of Its Underlying Graph

Matrix theory is a theory that studies applied mathematics.It has developed into an independent discipline due to as its extensive application in graph theory,algebra,combinatorial mathematics,statistics,etc.In order to characterize the structure of the graph,the researchers introduce special matrices,like adjacency matrix and skew-adjacency matrix,etc.In fact,the adjacency matrix of an undirected graph is a symmetric matrix,and the skew-adjacency matrix of the directed graph is an antisymmetric matrix.The Hermitian adjacency matrix of the mixed graph,what we interest in this paper,is the Hermitian matrix.As we all know,the Hermite matrix can be diagonalized,and the elements of the resulting diagonal matrix are real numbers.This means that the eigenvalues of the Hermitian matrix are real,and the rank of the Hermitian matrix is the same as the number of its non-zero eigenvalues.Therefore,the H-rank of the mixed graph equal to the multiplicity of non-zero eigenvalues.This paper discusses the relationship between the H-rank of the mixed graph and the matching number of the underlying graph.Moreover,we divide conditions of the mixed graphs that satisfies the upper bound or lower bound.The concrete content is in the following:· In Chapter 1.we give the definitions and symbols needed for this paper,briefly introduces the research background of the article and the research status of scholars at home and abroad.Moreover,it lists main problems and some corol?lary.· In Chapter 2,we give some important lemmas.-2d(G)≤rk(G)-2m(G)≤d(G)has been proved,from which we can deduce the conclusion in[25].· In Chapter 3,Firstly,by analyzing the coefficients of the characteristic poly-nomial of the mixed graph,the optimal conditions on the unicyclic graph are proved.By induction on|VG|to prove the theorem 1.13.The theorem in[20]has been included in this case.· In Chapter 4,Firstly,by analyzing the coefficients of the characteristic polyno-mial of the mixed graph,the lower conditions on the unicyclic graph are proved.By induction on|VG|to prove the theorem 1.14.The theorem in[22,14]has been included in this case.