Primitive Index And Application To The Graph

Graph theory is an emerging discipline and a branch of combinatorial mathematics. It has been widely applied in many different fields, such as physics, chemistry, operation research, computer science, information theory, cybernetics, network theory, social science as well as economical management. The matrix A can build correspondence relations with the concomitant directed graph D(A). So we can solve some nonnegative matrix problems using the knowledge of graph theory.Its primary coverage is:In chapter 1, firstly, we outline the history of development on graph theory. Secondly, we introduce some elementary knowledge and the domestic and foreign research survey of the primitive exponents of directed digraph .Thirdly, we give the Application of directed digraph .Lastly, we propose our research problems.In chapter 2, we consider the special two-colored digraphs D_n . it contains two cycles which consists of one 2m-cycle and one 2m + 1-cycle. All primitive conditions are proved. We find the bounds on the exponents by the inverse matrix. Further, we give the characterizationsof extremal two-colored digraphs and find the exponent set.In chapter 3, we consider the special two-colored digraphs D_(n,t,p) . It contains two cycles which consists of one (n + 1)-cycle and one (tn + t + 1)-cycles and has changeable public arcs Some primitive conditions are proved. We find the bounds on the exponents by the inverse matrix. Further, we give the bounds on the exponents and the characterizations of extremal two-colored digraphs in a special case.In chapter 4, we consider the application of directed digraph. We use two models to solve the problems , one is the Maximum Flow model, another is TSP model . We used the software of LINGO to solve the problems.