Multicolor Exponent, A Directed Graph

Combinatorial mathematics is also called combinatorial theory or combinatorial analysis , is a branch of mathematics. In daily life ,we always meet with many questions of combinatorial mathematics ,such as financial analysis , determination of investment programme, logistics planning, computer science, information theory, cybernetics as well as network algorithm and analysis.Graph theory and nonnegative matrix theory are two main research contents of combinatorial mathematics,this two contents have closer relationship . Nonnegative 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 select three classes of special directed digraphs with certain representation . In other word, we select a class of primitive two-colored digraphs with three cycles and a class of primitive two-colored digraphs with two double direction cycles for 2-primitive, and a class of primitive three-colored digraphs with three cycles for 3-primitive.Its primary coverage is:In chapter 1, firstly, we introduce the relational concepts of nonnegative matrix . Secondly, from the relation of graph and nonnegative matrix we introduce some elementary knowledge and the domestic and foreign research survey of the primitive matrixes and primitive exponents of directed digraph. Lastly, we propose our research problems.In chapter 2, we consider the special two-colored digraphs with tree cycles D whose cycle-lengths are n, n—1 and n—2 respectively. We discuss all coloring conditions and list its primitive conditions. We give the upper bounds on the exponents of all kinds of primitive conditions.In chapter 3, we consider the special two-colored digraphs with two double-direction cycles D.For it has two double-direction cycles,in fact it has four cycles and all of this four cycles are n-cycle .Samely ,we give the upper bounds on the exponents of all kinds of primitive conditions.In chapter 4, we consider the special three-colored digraphs D and it just consists three cycles whose cycle-lengths are n, n—2 and 3 respectively. Every primitive condition under the coloring condition is discussed . We find a upper bound on the exponent by the inverse matrix. Further, we give the characterizations of extremal three-colored digraphs.