The Iteration Digraphs Of Finite Commutative Rings

Let S be a finite commutative ring (resp. a finite abelian group). The iteration digraph G(S, k) of S, is a directed graph whose vertices are the elements in S, and for which there is a directed edge from α∈S to β∈S if and only if αk=β. In this thesis, we apply the ring theory, number theory and graph theory to investigate the iteration digraphs of finite commutative rings. The main contents are as follows.First of all, we summarize the background and review the research progresses of home and abroad of the related issues. Moreover, we state the contents and describe the approach of the present thesis. In other chapters we study the iteration digraphs of finite commutative rings R, finite abelian groups, the group algebra FqCn of cyclic groups Cn over finite fields Fq, and the ring Zn[i] of Gaussian integers modulo n, respectively. We establish necessary and sufficient conditions for G(R, k) to contain t-cycles, and obtain the length of the longest cycle in G(R, k) when k runs over positive integers. Additionally, by using Mobius inversion formula we determine the number of cycles in G1(R, k), the subgraph of G(R, k) induced by all the elements of the unit group of R. Moreover, we present the general results concerning the semiregularities and symmetries of G(R, k). In addition, for arbitrary finite abelian groups H, we explore the equivalent conditions for G(H,k1)=G{H,k2) or G{H,k1)≌G(H,k2), where k1and k2are two distinct positive integers. The fundamental constituents G*N(R,k) of G(R,k) are investigated and we obtain necessary and sufficient conditions for GN*(R, k) to be regular or semiregular. We also present some conditions when trees attached to cycle vertices in different fundamental constituents are isomorphic. Moreover, for the digraph G(FqCn,k), we characterize q, n and k for which the digraph is symmetric or semiregular. Finally, we completely determine the semiregularities of G2(Zn[i],3) via the indegree of some special vertices. And the heights of vertices and components in G(Zn[i],3) are established.