On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo

Let be the zero divisor graph for the ring of the Gaussian integers modulo . Several properties of the line graph of , are studied. It is determined when is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of is given when is a power of a prime. On the other hand, several graph invariants for are also determined. 1. Introduction The study of zero divisor graphs of commutative rings reveals interesting relations between ring theory and graph theory; algebraic tools help understand graphs properties and vise versa. In 1988, Beck [1] defined the concept of zero divisor graph of a commutative ring , where the vertices of this graph are all elements in the ring and two vertices , are adjacent if and only if . Anderson and Livingston [2] modified the definition of zero divisor graphs by restricting the vertices to the nonzero zero divisors of the ring . Further study of zero divisor graphs by Anderson et al. [3] investigated several graph theoretic properties, such as the number of cliques in . They also gave some cases in which is planer. On the other hand, they answered the question when for some specified types of rings and . Akbari and Mohammadian [4] improved on those results. for rings which satisfy certain conditions are discussed by Anderson and Badawi [5]. The zero divisor graph of the ring of integers modulo was extensively studied in [6–10]. In 2008, Abu Osba et al. [11] introduced the zero divisor graphs for the ring of Gaussian integers modulo , , where they studied several graph properties and determined several graph invariants for . Further properties of the zero divisor graphs for the ring of Gaussian integers modulo are investigated in [12]. In this paper, we study the line graph of . We organized our work as follows: some basic definitions and terminology are given in Section 2. In Sections 3 and 4, we answer the question when is the line graph Eulerian, Hamiltonian, or planer. In Section 5, the chromatic and clique numbers of are found. While the diameter, the girth and the radius of are determined in Sections 6 and 7, respectively. Finally, the last two sections discuss the domination number of and as well as the independence and clique numbers of . 2. Preliminaries The set of Gaussian integers is defined by . A prime Gaussian integer is one of the following:(i) or ,(ii) , where is a prime integer and ,(iii) , , where , is a prime integer and . It is clear that is a ring with addition and multiplications modulo .