Centrosymmetric Primitive Exponents Of Primitive Matrices

In this paper, the exponents of central symmetric primitive matricesis studied. We use graph-theoretic version to relate, use graph-theoreticmethods and techniques to prove our results. The associated digraph of acentral symmetric primitive matrice is a primitive directed digraph. Inthis paper, we prove that the maximum of the exponents of centralsymmetric primitive matrices is n-1, we also characterize the exponentset of central symmetric primitive matrices. In addition, the gap of theexponents of central symmetric primitive matrices with the length d ofits minimum odd circle is concluded, we also characterize the matriceswhose exponents are the second larger number.In chapter 1, we introduce a few basic concepts, and summarize thebackground and advancement of the research on exponents.In chapter 2, we prove that the maximum of the exponents of centralsymmetric primitive matrices is n-1, we also characterize the exponentset of central symmetric primitive matrices.In chapter 3, we study the exponents of central symmetric primitivematrices with the length d of its minimum odd circle, and conclude someresults as the following:(1) In 3.1, some gap of the exponents of central symmetricprimitive matrices(n=2m+1) with the length d of its minimum odd circleis characterized, that is {n-d+1,…,n-2}(?)(n, d).(2) In 3.2, some gap of the exponents of central symmetricprimitive matrices(n=2m) with the length d of its minimum odd circle ischaracterized, that is {n-d+1,…,n-2}(?)(n,d).In chapter 4, we characterize the matrices whose exponents are thesecond larger number.