On The Multi-color Primitive Exponents Of The Directed Graph

Combinatorial mathematics is a branch of mathematics. In daily life, we always meetwith many questions of combinatorial mathematics, such as financial analysis, determinationof investment programme, logistics planning, computer science, information theory cyber-netics as well as network algorithm and analysis. Graph theory and nonnegative matrixtheory are two main research contents of combinatorial mathematics this two contents havecloser relationship. Non-negative matrix A can build correspondence relations with the con-comitant directed graph D(A), so we can solve some nonnegative matrix problems usingthe knowledge of graph theory. In this article, we consider a class of primitive three-coloreddigraphs with odd vertices, and a class of special primitive three-colored digraphs with threecycles.In chapter 1, firstly, we introduce the relational concepts of nonnegative matrix. Sec-ondly, from the relation of graph and nonnegative matrix we introduce some elementaryknowledge and the domestic and foreign research survey of the primitive matrixes and prim-itive exponent of directed digraph. Lastly, we propose our research problems.In chapter 2, we consider a class of primitive three-colored digraphs with odd verticeswhose cycle-lengths are n, (n ? 2) and 2 respectively. We discuss all coloring conditions andlist its primitive conditions. We give the upper bounds on exponents of all kinds of primitivecondition by the inverse matrix and we give the characterizations of extremal second-coloreddigraphs. Finally we give a special example.In chapter 3, we consider a class of special primitive three-colored digraphs with oddvertices whose cycle-lengths are n, 3 and 4 respectively. We discuss all coloring conditionsand list its primitive conditions. We give the upper bounds on exponents of all kinds ofprimitive condition by the inverse matrix. Further we give the characterizations of extremalsecond-colored digraphs.