Primitive Almost About Matrix K-index

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 associated digraph of a primitive nearly reducible matrix is a primitive minimally strong digraph.In 1982,J.A.Ross characterized the maximum and extremal digraph of the exponent of primitive minimally strong with n vertrices and grith g.In 1991,Jiaoyu Shao characterized the exponent set.In 1999,Bolian Liu characterized the maximum.In 2002,Bo Zhou characterized the extremal digraph.In 2005,Yahui Hu generalize the results of J.A.Ross to the case of the k-exponent and completely characterize the 1-exponent set.On those basis,we know that the primitive minimally strong with n vertrices and grith 2.In this thesis we completely characterize the 1-exponent of the primitive minimally strong with n vertrices and grith 2.In Chapter 1,we introduce a few basic concepts,summarize the background and advancement of the research on generalized exponents.In Chapter 2,we introduce the exponent of the vertex and some basic knowledge about the primitive minimally strong digraph,we mainlly introduce the Frobenius,the numberexp_D(u),the number exp_D(u,v)and it's estimation.Last we introduce some result about the primitive minimally strong digraph and the maximum and extremal digraph of the primitive minimally strong digraph.In Chapter 3,1-exponents of primitive minimally strong digraphs are studied when n is even.The following resuits are obtained: E_n(1) = {4,5,6,7,8,9,10,11…2n-7,2n-6,2n-5,2n-4} where n is even.In Chapter 4,1-exponents of primitive minimally strong digraphs are studied when n is odd.The following resuits are obtained: E_n(1) = {4,5,6,7,8,9,10,11…2n-8,2n-7,2n-6,2n-5} where n is odd.