Graph Operations For The Matrix Of The Primitive Index And Matrix Operations

The combinational theory of nonnegative matrix researches the qualities that depend on the pattern of matrix and to be unconcerned with the value of matrix element.It has close relation with some quality of graph,and has relatively application in many areas such as information science, communication networks, computer science.Primitive exponent and generalized primitive exponent of primitive matrices are important research content in the combinational theory of nonnegative matrix.So far, many problems have been resolved. In the new context,the research for primitive exponent of nonnegative matrix pairs came into being.In fact, there is a one-to-one relationship between nonnegative matrix pairs and two-colored digraph,so the problem of matrices can transform into the problem of graphics to solve.This paper studies one classes of special two-colored digraph, the main contents as follows:In chapter 1, firstly the development and content on graph theory and combinational theory of nonnegative matrix are described roughly. Then, some elementary concepts and the domestic and foreign research survey of the primitive exponents of directed digraph are introduced. Lastly, our research problems are proposed.In chapter 2,a special two-colored digraph with two cycles is considered, whose uncolored digraph consists of one((t—1)m + 1)-cycle and one(tm + 1)-cycle.Using the method of combinatorial matrix theory and graph theory, the primitivity conditions and the bound on the exponent are given.Further,the exponent set is obtained and the extremal digraphs that meet the upper bound and the lower bound are characterized.In chapter 3,use the connection of n-square matrix and its weighted digraph,put forward a graphical method to solve the operation of matrix,and use this method in chapter 2.