The Generalized Exponents Of Primitive Nearly Reducible Matrices

This thesis is devoted to the study of the k -exponent of primitive, nearly reducible matrices. We use graph-theoretic version to relate, use graph-theoretic methods and techniques to prove our results. The asso -ciated digraph of a primitive nearly reducible matrix is a primitive minimally strong digraph. In 1982, J. A. Ross[1] characterized the maximum exp(PMSD_n,g,n) and extremal digraphs exp(PMSD_n,g,n) of the exponent of primitive minimally strong digraphs with n vertices and girth g. In 1991, Jiayu Shao[2] characterized the exponent set (namely the n -exponent set) exp(PMSD_n,n). In 1999, Bolian Liu[3] characterized the maximum exp(PMSD_n,k). In 2002, Bo Zhou[4] characterized the extremal digraphs exp(PMSD_n,k) . However, the k-exponent sets exp(PMSD_n,k) (1≤k≤n-1)are not characterized yet. In 2000, Zhengke Miao[5] suggested that the complete determination of exp(PMSD_n,k) is an open problem in his Ph. D. Degree Thesis. Bo Zhou[4] pointed out that the complete determination of exp(PMSD_n,k) is an interesting and difficult problem. In this thesis, we generalize the results of [1] to the case of the k-exponent, and completely characterize exp(PMSD_n,1).In Chapter 1, we introduce a few basic concepts, summarize the background and advancement of the research on generalized exponents.In Chapter 2, the k-exponents of primitive minimally strong digraphswith n vertices and the girth g are studied. We obtain the maximum exp(PMSDng,k) and the extremal digraph exp(PMSDng,k). By this result, we can obtain Qxp(PMSDn,k) and &q>(PMSDn,k) easily.In Chapter 3, 1-exponents of primitive minimally strong digraphs are studied. The following results are obtained:(1) In Section 3. 1, we study the upper bound of 1-exponents of primitive minimally strong digraphs with n vertices and not less than three lengths of cycles. We prove the result that if ?>14 and \L(D)\>3, then expD (1) < -(n2 - In+16).(2) In Section 3. 2, we define a series of new concepts such as the consecutive p-cycle, the consecutive p-cycles cover and the consecutive p-cycles chain, and so on. By studying their properties and a few new properties of primitive minimally strong digraphs, we obtain the lower bound of 1 -exponents of the primitive minimally strong digraph D with n vertices and L(D) = {p,q} (3n).(3) In 3. 3, we characterize the 1-exponent set of the primitive minimally strong digraphs D with n vertices and L(D) = {p,q} Qn).(4) In 3. 4, we prove that if ?>14, then Vwe[4,...,-(?2-7?+l6)], there exists a digraph D&PMSD? such that expD(l) = /M.(5) In 3. 5, we obtain the result that expD(l)>4 for D&PMSDn, and we completely characterize exp(PMSDn,l) and exp(PMSDf\l):where6n\p-2f+ 9-21 -— >n /,2|...