Hits:

Indexed by:期刊论文

Date of Publication:2010-10-01

Journal:ARS COMBINATORIA

Included Journals:SCIE、Scopus

Volume:97

Page Number:101-110

ISSN No.:0381-7032

Key Words:Cubic mapping graph; Carmichael lambda-function; Chinese remainder theorem; Component of a graph

Abstract:In this paper, we study the connection of number theory with graph theory via investigating some uncharted properties of the directed graph Gamma(n) whose vertex set is Z(n) = {0, 1, ..., n - 1}, and for which there is a directed edge from a is an element of Z(n) to b is an element of Z(n) if and only if a(3) equivalent to b (mod n). For an arbitrary prime p, the formula for the decomposition of the graph F(p) is established. We specify two subgraph Gamma(1)(n) and Gamma(2)(n) of F(n). Let Gamma(1)(n) be induced by the vertices which are coprime to n and Gamma(2)(n) by induced by the set of vertices which are not coprime to n. We determine the level of every component of Gamma(1)(n), and establish necessary and sufficient conditions when Gamma(1)(n) or Gamma(2)(n) has no cycles with length greater than 1, respectively. Moreover, the conditions for the semiregularity of Gamma(2)(n) are presented.