Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say 􀀀 (RM), such that when M = R, 􀀀 (RM) coincide with the zero-divisor graph of R. Many well-known results by D. F. Anderson and P. S. Livingston, have been generalized for 􀀀 (RM). We will show that 􀀀 (RM) is connected with diam(􀀀 (RM))  3, and if 􀀀 (RM) contains a cycle, then gr(􀀀 (RM))  4. We will also show that 􀀀 (RM) = ∅ if and only if M is a prime module. Among other results, it is shown that for a reduced module M satisfying DCC on cyclic submodules, gr (􀀀 (RM)) = 1 if and only if 􀀀 (RM) is a star graph. Finally, we study the zero-divisor graph of free R-modules.

Let R be a commutative ring with identity and M be an R-module. The zero divisor graph of M is denoted by 􀀀 (M). In this study, we are going to generalize the zero divisor graph 􀀀 (M) to Submodule-based zero divisor graph 􀀀 (M, N) by replacing elements whose product is zero with elements whose product is in some submodule N of M. The main objective of this paper is to study the interplay of the properties of submodule N and the properties of 􀀀 (M, N).

In this paper, we introduce the zero-divisor associate graph $\Gamma_D(R)$ over a finite commutative ring $R$. It is a simple undirected graph whose vertex set consists of all non-zero elements of $R$, and two vertices $a, b$ are adjacent if and only if there exist non-zero zero-divisors $z_1, z_2$ in $R$ such that $az_1=bz_2$. We determine the necessary and sufficient conditions for connectedness and completeness of $\Gamma_D(R)$ for a unitary commutative ring $R$. The chromatic number of $\Gamma_D(R)$ is also studied. Next, we characterize the rings $R$ for which $\Gamma_D(R)$ becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as $\Gamma_D(R)$, characterizing the associated ring $R$ in each case.

Let L be a lattice with the least element 0. An element x ϵ L is a zero divisor if x ^ y= 0 for some y ϵÎ L*= L\ {0}. The set of all zero divisors is denoted by Z (L). We associate a simple graph G(L) to L with vertex set Z (L)*= Z (L)\ {0}, the set of non-zero zero divisors of L and distinct x, y ϵ Z (L)* are adjacent if and only if x ^ y= 0. In this paper, we obtain certain properties and diameter and girth of the zero divisor graph G(L). Also we find a dominating set and the domination number of the zero divisor graph G(L).

For a commutative semigroup S with 0, the zero-divisor graph of S denoted by G(S) is the graph whose vertices are nonzero zero-divisor of S, and two vertices x, y are adjacent in case xy = 0 in S. In this paper we study median and center of this graph. Also we show that if Ass(S) has more than two elements, then the girth of G (S) is three.

Let $R$ be a finite commutative ring with or without unity and $\Gamma_{e}(R)$ be its extended zero-divisor graph with vertex set $Z^{*}(R)=Z(R)\setminus \lbrace0\rbrace$ and two distinct vertices $x,y$ are adjacent if and only if $x.y=0$ or $x+y\in Z^{*}(R)$. In this paper, we characterize finite commutative rings whose extended zero-divisor graph have clique number $1 ~ \text{or}~ 2$. We completely characterize the rings of the form $R\cong R_1\times R_2 $, where $R_1$ and $R_2$ are local, having clique number $3,~4~\text{or}~5$. Further we determine the rings of the form $R\cong R_1\times R_2 \times R_3$, where $R_1$,$R_2$ and $R_3$ are local rings, to have clique number equal to six.

Let M be an R-module and 0¹¦ÎM*=Hom (M, R). The graph  G¦ (M) is a graph with vertices Z¦ (M)={x ÎM \ {0} |x¦(y) = 0 or y¦(x) = 0 for some non-zero yÎM}, in which non-zero elements x and y are adjacent provided that x¦(y) = 0 or y¦ (x)=0, which introduced and studied in [3]. In this paper we associate an undirected submodule based graph G¦N (M) for each submodule N of M with vertices Z¦N (M)= {xÎM \ N | x¦(y) ÎN or y¦ (x) ÎN for some y Î M \N}, in which non-zero elements x and y are adjacent provided that x¦ (y)ÎN or y¦(x) ÎN. We observe that over a commutative ring R, G¦N (M) is connected and diam (G¦N (M)) £3. Also we get some results about clique number and connectivity number of G¦ N (M).