‎Let $R$ be commutative ring with identity and $M$ be an $R$-module‎. ‎The zero divisor graph of $M$ is denoted $Gamma{(M)}$‎. ‎In this study‎, ‎we are going to generalize the zero divisor graph $Gamma(M)$ to submodule-based zero divisor graph $Gamma(M‎, ‎N)$ by replacing elements whose product is zero with elements whose product is in some submodules $N$ of $M$‎. ‎The main objective of this pa...

Let $G=(V,E)$ be a simple graph. A set $Ssubseteq V$ isindependent set of $G$,  if no two vertices of $S$ are adjacent.The  independence number $alpha(G)$ is the size of a maximumindependent set in the graph. In this paper we study and characterize the independent sets ofthe zero-divisor graph $Gamma(R)$ and ideal-based zero-divisor graph $Gamma_I(R)$of a commutative ring $R$.

For a commutative semigroup S with 0, the zero-divisor graph of S denoted by &Gamma(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 &Gamma(S) is three.

the concept of the bipartite divisor graph for integer subsets has been considered in [‎graph combinator., ‎ 26 (2010) 95--105‎.]. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g$ is a solvable group‎, ‎then the number of connected components of this graph is at most $2$ and if $g...

let $g=(v,e)$ be a simple graph. a set $ssubseteq v$ isindependent set of $g$,  if no two vertices of $s$ are adjacent.the  independence number $alpha(g)$ is the size of a maximumindependent set in the graph. in this paper we study and characterize the independent sets ofthe zero-divisor graph $gamma(r)$ and ideal-based zero-divisor graph $gamma_i(r)$of a commutative ring $r$.

‎the concept of the bipartite divisor graph for integer subsets has been considered in [‎m‎. ‎a‎. ‎iranmanesh and c‎. ‎e‎. ‎praeger‎, ‎bipartite divisor graphs for integer subsets‎, ‎{em graphs combin.}‎, ‎{bf 26} (2010) 95--105‎.]‎. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g...

let $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...

‎in this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $c_4$-free.‎

Let $R$ be a commutative ring and $I$ an ideal of $R$. The zero-divisor graph of $R$ with respect to $I$, denoted by $Gamma_I(R)$, is the simple graph whose vertex set is ${x in Rsetminus I mid xy in I$, for some $y in Rsetminus I}$, with two distinct vertices $x$ and $y$ are adjacent if and only if $xy in I$. In this paper, we state a relation between zero-divisor graph of $R$ with respec...