On The Relation Between The Adjacency Rank Of A Complex Unit Gain Graph And The Matching Number Of Its Underlying Graph

The research on establishing the relationship between spectral parameters of graph-matrices and the structure theory(resp.structure parameters)of graphs is a hot topic in the study of graph theory.To promote this basic theoretical research can not only improve the original innovation ability but also provide mathemati-cal supports for the development of the theoretical physics,quantum chemistry and theoretical computer science.Let Gφ be an n-vertex complex unit gain graph and let G be its underlying graph.The adjacency rank of Gφ,written as r(Gφ)is the rank of its adjacency matrix and denote by α’(G)the matching number of the underlying graph G.In this paper,the relationship between the T-gain graph Gφ and the matching number of its underlying graph is characterized.The concrete content is in the following:·In Chapter 1,we firstly introduce the research background of this article and the existed research results,then we list our main research results.·In Chapter 2,some fundamental symbol and concept,and some important lemmas are introduced.·In Chapter 3,based on combinatorial interpretation of all the coefficients of the characteristic polynomial of we determine sharp upper and lower bounds on r(Gφ)-2α’(G),all extremal unit gain graphs are characterized.Furthermore,we establish sharp lower bounds on the difference and quotient between the adjacency rank of Gφ and the matching number of its underlying graph.And the corresponding extremal complex unit gain graphs are determined.·In Chapter 4,we summarize the principal conclusions in this paper and make some prospects for further study.