The K -point Exponent Of Doubly Symmetric Primitive Matrices With Non-zero Trace

Combinatorial matrix theory is an important field of combinatorial mathematics.It has close contaction with graph theory, number theory , linear algebra and probability and statistics and other branches of mathematics , and it has been widely applied in communication network theory, computer science, sociology, biology and economics .In 1990, Brualdi and bolian Liu introducted the generalized exponents of Boolean matrix ( directed graphs) in the backgroud of memoryless communication systems. They are the generalization of the traditional primitive exponent .The upper bounds, the sets and the extermal matrices of the generalized exponents are three important problems in the study. The generalized exponents contain k-point exponent,the kth upper multiexponent and the kth lower multiexponent. In this paper we will study the k-point exponent of doubly symmetric primitive matrices with non-zero trace with the graph theory, the following major contents are: In the first chapter we mainly explain the research background of the generalized exponents,some basic concepts and the current research situations; In the second chapter we prove the upper bounds of k-point exponent of doubly symmetric primitive matrices with non-zero trace; In the third chapter we first construct two types of graphs, and then charact the extermal matrices of the k-point exponent of doubly symmetric primitive matrices with non-zero trace;In the fourth chapter we discuss the k-point exponent sets of doubly symmetric primitive matrices with non-zero trace; In the fifth chapter we point out the contents that will be futher studied.