Matrix Cluster Exponent,

Graph theory is a branch of combinatorial mathematics. It has been widely appliedin many di?erent fields, such as physics, chemistry, operation research, computer science,information theory, cybernetics, network theory, social science as well as economical man-agement. The matrix A can build correspondence relations with the concomitant directedgraph D(A). So we can solve some nonnegative matrix problems using the knowledge ofgraph theory. In this article, we consider two classes of primitive three-colored digraphs. Itsprimary coverage is :In chapter 1, firstly, we outline the history of development on graph theory. Secondly,we introduce some elementary knowledge of graph theory and the domestic and foreignresearch survey of the primitive exponents of directed digraph. Lastly, we propose ourresearch problems.In chapter 2, we consider the special three-colored digraphs D whose uncolored digraphconsists of one n-cycle, one (n ? 1)-cycle and one 2-cycle. Some primitive conditions areproved. We find a upper bound on the exponent by the primitive inverse matrixIn chapter 3, we consider the three-colored digraphs D whose uncolored digraph consistsof one n-cycle, one (n ? 1)-cycle and one 3-cycle. Some primitive conditions are proved. Wefind a upper bound on the exponent by the primitive inverse matrix.