Study Of Three Types Of Generalized Primitive Exponents Of Primitive Matrices

Combinatorial matrix theory is an important field of combinatorial mathematics, it is closely associated with the graph theory, the number theory, linear algebra, and probability statistics; and it also be used widely in communication network theory, computer science, sociology, and economics. In1990, Brualdi and Liu B.L. on the background of memoryless communication system, introduced the generalized primitive exponents of boolean matrices, which is the extend of primitive exponents. The upper bound of generalized primitive exponents and the definition of corresponding indices sets are the important content of indices study. Generalized exponents include k-th generalized exponents, k-th upper generalized exponents, and k-th lower generalized exponents. The main content of this article summarizes the development of combinatorial matrix theory, introduces some basic knowledge of it, and three kinds of generalized exponents of boolean matrix. We briefly introduce the study of the first type generalized exponents and the second type generalized exponents, after that, we focus on the study of the third type generalized exponents, and get the maximum of the third generalized exponents, describe its pole matrix. We also organize and determine the exponent set.