Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n

Osama Alkam,Emad Abu Osba

doi:10.1155/2014/146873

Abstract

Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

Highlights

Let ω be a primitive third root of unity
In this paper we study the zero divisor graph Γ(En)
We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal

Summary

Introduction

Let ω be a primitive third root of unity. the set of complex numbers a + bω, where a, b are integers, is called the set of Eisenstein integers and is denoted by E. Since En is element of En a is finite commutative ring a unit or a zero divisor. With identity, every Let |Z∗(R)| denote the number of nonzero zero divisors of a ring R. An earlier study was carried out for the zero divisor graph Γ(Zn[i]) of the ring of Gaussian integers modulo n (see [8]). The girth g(Γ(R)) of the graph Γ(R) is ∞ if the graph contains no cycles; otherwise it is the length of the shortest cycle It is shown in [4] that, for any commutative ring with identity, there is a path between any two vertices of Γ(R) and that diam(Γ(R)) ≤ 3. Throughout this paper we will use p to denote a usual prime integer that is congruent to 2 modulo 3 and use q to denote other prime integers

Diameter and Girth of the Zero Divisor Graph of Eqk

Diameter and Girth