Some digraphs arising from number theory and remarks on the zero-divisor graph of the ring <i>Z_n</i>

作     者：Skowronek-Kaziow, Joanna 

作者机构：Univ Zielona Gora Fac Math PL-65516 Zielona Gora Poland 

出 版 物：《INFORMATION PROCESSING LETTERS》 (信息处理快报)

年 卷 期：2008年第108卷第3期

页      面：165-169页

核心收录：

学科分类：08[工学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

主　　题：Digraph Chinese remainder theorem Carmichael lambda-function Group theory Zero-divisor graph Graph algorithms 

摘      要：In the first part of the paper we investigate a digraph Gamma(n) whose set of vertices is the set H = {0, 1,..., n - 1} and for which there is a directed edge from a is an element of H to b is an element of H if a(3) equivalent to b (mod n). We specify two subdigraphs Gamma(1)(n) and Gamma(2)(n) of Gamma(n). Let Gamma(1)(n) be induced by the vertices which are coprime to n and Gamma(2)(n) be induced by the set of vertices which are not coprime with n. The conditions for regularity and semiregularity of these subdigraphs are presented. The digraph F(n) has an interesting structure for some special n. It is shown that every component of the digraph Gamma(n) is a cycle if and only if 3 does not divide the Euler totient function phi(n) and n is square-free. It is proved that Gamma(1)(2(k)) contains only cycles and Gamma(2)(2(k)) is a tree with the root in 0. Besides Gamma(1)(3(k)) contains two ternary trees with roots in 1 and 3(k) - 1 and Gamma(2)(3(k)) is a tree with the root in 0. All digraphs with 3 components are described. In the second part we consider the zero-divisor graph G(Z(n)) of the ring Z(n). Its maximum degree and the clique number are calculated. (C) 2008 Elsevier B.V. All rights reserved.