Matrix Cluster Exponent,

The combinational theory of nonnegative matrix is on the rise since the 1950th anda rapidly developing branch of mathematics. It researches the qualities that depend onthe pattern of matrix and to be unconcerned with the value of matrix element. It hasnot only an extremely important position at basic math study but also have importantapplications in other disciplines.For instance, there is a specific application background ininformation science, communication networks, computer science, coding and cryptography,physics, chemistry, biology, sociology and many other aspects of the economy .Primitive exponent and generalized primitive exponent of primitive matrices are im-portant research content in the combinational theory of nonnegative matrix.So far, manyproblems have been resolved. In the new context,the research for primitive exponent of non-negative matrix pairs and tuples of nonnegative matrices came into being.In fact, there is aone-to-one relationship between nonnegative matrix pairs and two-colored digraph, and thesame as tuples of nonnegative matrices and k-colored digraph, so the problem of matricescan transform into the problem of graphics to solve.This paper studies a class of specialtwo-colored digraph and a class of (k+1)-colored digragh, the main contents as follows:In chapter 1, firstly the development and content on graph theory and combinationaltheory of nonnegative matrix are described roughly. Then, some elementary concepts andthe domestic and foreign research survey of the primitive exponents of directed digraph areintroduced. Lastly, our research problems are proposed.In chapter 2,a special two-colored digraph with two cycles is considered,whose uncoloreddigraph consists of one n?cycle and one (n - t)-cycle. The primitivity conditions and theupper bound on the exponent are given.Further, the extremal digraphs that meet the upperbound are characterized.In chapter 3,a class of (k+1)-colored digragh are discussed. Combining with the Zna′mproblem in number theory, with the application of combinatorial matrix theory and graphtheory, the algorithms about the primitive exponents of one-arc light digraphs are given, and some primitive exponents of one-arc light digraphs are listed.