Zero Divisor Graph for the Ring of Eisenstein Integers Modulo

Let be the ring of Eisenstein integers modulo . In this paper we study the zero divisor graph . We find the diameters and girths for such zero divisor graphs and characterize for which the graph is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal. 1. Introduction Let be a primitive third root of unity. Then the set of complex numbers , where are integers, is called the set of Eisenstein integers and is denoted by . Since is a subring of the field of complex numbers, it is an integral domain. Moreover, the mapping is a Euclidean norm on . Thus is a principal ideal domain. The units of are , , and . The primes of (up to a unit multiple) are the usual prime integers that are congruent to modulo and Eisenstein integers whose norm is a usual prime integer. It is easily seen that, for any positive integer , the factor ring is isomorphic to the ring . Thus is a principal ideal ring. This ring is called the ring of Eisenstein integers modulo . In [1] this ring is studied and its properties are investigated; its units are characterized and counted. Thus, its zero divisors are completely characterized and counted. This characterization uses the fact that is a unit in if and only if is a unit in . Recall that a ring is local if it has a unique maximal ideal. The following are sample results of [1].(1)If is a prime integer, then the ring is local if and only if or .(2)Let denote the number of units in a ring ; then(i);(ii).We deduce the following. Proposition 1. Let , where , , , , and are primes such that , , for each . Then Proof. If , then . Thus, . Since is a finite commutative ring with identity, every element of is a unit or a zero divisor. Let denote the number of nonzero zero divisors of a ring . Then(1); (2) if ;(3) if ;(4)if , then The (undirected) zero divisor graph of a commutative ring with identity that has finitely many zero divisors is the graph in which the vertices are the nonzero zero divisors of . Two vertices are adjacent if they are distinct and their product is . The concept of a zero divisor graph was introduced by Beck in [2] and then studied by Anderson and Naseer in [3] in the context of coloring. The definition of zero divisor graphs in its present form was given by Anderson and Livingston in [4]. Numerous results about zero divisor graphs were obtained by Akbari et al. (see [5–7]). The zero divisor graph is studied to get a better understanding of the algebraic structure of the ring . The interplay of the algebraic properties of , graph theoretic properties of , and its relation with is studied. An