Several Types Of Color And Three-color Exponent, A Directed Graph

Graph theory is 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. In this article, we consider a class of primitive two-colored digraphs with two cycles, a class of primitive two-colored digraphs with three cycles 2-primitive, and a class of primitive three-colored digraphs with three cycles 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 theprimitive exponents of directed digraph. Lastly, we propose our research problems.In chapter 2, we consider the special two-colored digraphs D whose uncolored digraph has m + n vertices and consists of one m-cycle and one n-cycle, where m > n. Some primitive conditions are proved. We find a upper bound on the exponents by the inverse matrix. Further, we give the characterizations of extremal two-colored digraphs.In chapter 3, we consider the special n orders two-colored digraphs D whose consists of one n—cycle and two (n—2)—cycles. All of primitive conditions are given by discussing all coloring of six non-common arcs. Further, we give a upper bound on the exponents and the characterizations of extremal two-colored digraphs.In chapter 4, we consider the special three-colored digraphs D whose uncolored digraph consists of one n-cycle, one (n—1)-cycle and one 2-cycle. Some primitive conditions are proved. We find a upper bound on the exponent by the primitive inverse matrix. Further, we give the characterizations of extremal three-colored digraphs.