Types Of Matrix Cluster Exponent,

Graph theory and nonnegative matrix theory are two main research contents of combinatorial mathematics, this two contents have closer relationship.Nonnegative matrix A can build correspondence relations with the concomitant directed graph D(A),and the primitivity of colored multidigraphs is related to properties of the Hurwitz produce of matrices.A multidigraph D with arcs colored by c₁,c₂,…,c_k(k = 1,2,3,…) is primitive if there exists a k-tupleα-(α₁,α₂,…,α_k) of nonnegative integers such that for any u, v in V(D),there exists a (u,v)-walk which contains exactlyα_i arcs of color c_i for i = 1,2,…,k.If D is primitive,the exponent of D,denoted by exp(D) ,is the smallest value ofα₁ +α₂ +…+α_k over all suchα.The aim of this paper is to explore the primitivity and exponents of several classes of colored mutidigraphys. The paper divides into four chapter.In Chapter 1,after a short introduction to the used basic notation and terminology on graphs,from the relation of graphs and nonnegative matrix, we give some elementary knowledge and the domestic and foreign research survey of the primitive matrixes and primitive exponents of digraph.Lastly,we propose our research problems.In Chapter 2,we introduce primitivity and exponents of the special two-colored Wielandt digraph ,and we also introduce the exponents of a nonnegative matrix pair whose associated digraph of two cycles.In Chapter 3,we define a class of the special three-colored digraphs with odd vertices ,whose uncolored digraph consists of one n-cycle,one (n-2)-cycle and one 2-cycle.The exponent of compact supper bound and the characterization of the extremal for one three-colored primitive digraph is given.In Chapter 4,we introduce a class of multidigraphs D^* which consist of (k+ 1)-cycle,and is colored by c₁,c₂,…,c_k(k=1,2,3,…),and show a necessary and sufficient condition for it to be primitive.In Section 4.2,we summarize the content of this paper ,and propound the further job prospect.