Promotion Of The Primitive Exponents Of The Directed Graph With A Spectrally Arbitrary Pattern

Combinatorial matrix theory is an important branch of combinatorial mathematics.Its main research content is combinatorial nature of matrix relating with the symbol ofmatrix elements only and unrelating with the size of matrix elements. The combinatorialnature is very colse to some nature of graphs with specific application in information science,communication network, computer sicence, and so on. In this paper, we study generalizationsof primitive exponents of digraphs and spectrally arbitrary patterns, which are all importantresearch topics in combitorial matrix theory.In chapter 1, we outline the development and the significance on the primitive exponents,research survey of spectrally arbitrary patterns and main research work in this paper.In chapter 2, we mainly consider a class of two-colored digraphs D_n,s with many cycles.Firstly, we classify the digraphs by coloring. secondly, we study the primitivity conditionsof all kinds of digraphs. Lastly, we give upper bound on the exponents of D_n,s and thecharacterizations of extremal two-colored digraphs.In chapter 3, we study the primitivity condition and the primitive exponents of a classof infinite digraph D_i,j^k by using the local exponents of two vertices. At last, we present theprimitive exponent sets of D_i,j^k .In chapter 4, we find a class of minimal spectrally arbitrary pattern K_n with 2n nonzeroentries by Nilpotent?Jacobian method. And we prove that every superpattern of K_n is aspectrally arbitrary pattern.