Exponents Of Two-colored Digraphs

The nonnegative matrix combinatorial theory studies the property that only rely on the matrix zero position pattern, but has nothing to do with number value of element. It has the close relation with certain property of graph. It has concrete application background in many respects such as information science, communication network, computer science, code and cryptology, physics, chemistry, biology, sociology, economic mathematics and so on. Since the eighties of last century, research on exponents and generalized exponents of primitive digraphs (primitive matrix) has made a rapid progress. A lot of problems have already been solved satisfactorily. Primitive exponents of two-colored digraphs (nonnegative matrix pairs) are generalizations of traditional primitive exponents of primitive digraphs. It is a brand-new research content, and has the important application in discrete homogeneous two-dimensional dynamical system. The determination of the bounds on the exponents and the set of exponents, and the characterizations of the extremal two-colored digraphs is an important problem in the study of primitive exponents of two-colored digraphs. In this paper, we according to one-to-one correspondence between nonnegative matrix pairs and two-colored digraphs, use knowledge of combinatorial mathematics, use graph-theoretic version to describe, use graph-theoretic methods and techniques to study a special primitive two-colored digraphs whose uncolored digraph has 2n+1 vertices, and consist of one (2n+1)-cycle and one (n+1)-cycle. We give the bounds on the exponents, the exponent set and the characterizations of the extremal two-colored digraphs. The article divides into three chapters.In chapter 1, we introduce the history of development on the primitive exponents, some basic knowledge, and our research problems and main results.In chapter 2, we discuss a special primitive two-colored digraphs, we give the bounds on the exponents, the exponent set and the characterizations of the extremal two-colored digraphs. In chapter 3, we summarize the content of this text, and propound the further job prospect.