1. CMB 2017 (vol 60 pp. 319)
 Khojasteh, Sohiela; Nikmehr, Mohammad Javad

The Weakly Nilpotent Graph of a Commutative Ring
Let $R$ be a commutative ring with nonzero identity. In this
paper, we introduced the weakly nilpotent graph of a commutative
ring. The weakly nilpotent graph of $R$ is denoted by $\Gamma_w(R)$
is a graph with the vertex set $R^{*}$ and two vertices $x$ and
$y$ are adjacent if and only if $xy\in N(R)^{*}$, where $R^{*}=R\setminus\{0\}$
and $N(R)^{*}$ is the set of all nonzero nilpotent elements
of $R$. In this article, we determine the diameter of weakly
nilpotent graph of an Artinian ring. We prove that if $\Gamma_w(R)$
is a forest, then $\Gamma_w(R)$ is a union of a star and some
isolated vertices. We study the clique number, the chromatic
number and the independence number of $\Gamma_w(R)$. Among other
results, we show that for an Artinian ring $R$, $\Gamma_w(R)$
is not a disjoint union of cycles or a unicyclic graph. For Artinan
ring, we determine $\operatorname{diam}(\overline{\Gamma_w(R)})$. Finally, we
characterize all commutative rings $R$ for which $\overline{\Gamma_w(R)}$
is a cycle, where $\overline{\Gamma_w(R)}$ is the complement
of the weakly nilpotent graph of $R$.
Keywords:weakly nilpotent graph, zerodivisor graph, diameter, girth Categories:05C15, 16N40, 16P20 

2. CMB 2016 (vol 59 pp. 794)
 Hashemi, Ebrahim; Amirjan, R.

Zerodivisor Graphs of Ore Extensions over Reversible Rings
Let $R$ be an associative ring with identity.
First we prove some results about zerodivisor graphs of reversible
rings. Then we study the zerodivisors of the skew power series
ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$compatible. Moreover, we compare the diameter and girth of the zerodivisor
graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$,
when
$R$ is reversible and $(\alpha,\delta)$compatible.
Keywords:zerodivisor graphs, reversible rings, McCoy rings, polynomial rings, power series rings Categories:13B25, 05C12, 16S36 

3. CMB 2015 (vol 58 pp. 271)
4. CMB 2014 (vol 57 pp. 573)
 Kiani, Sima; Maimani, Hamid Reza; Nikandish, Reza

Some Results on the Domination Number of a Zerodivisor Graph
In this paper, we investigate the domination, total domination and
semitotal domination numbers of a zerodivisor graph of a
commutative Noetherian ring. Also, some relations between the
domination numbers of $\Gamma(R/I)$ and $\Gamma_I(R)$, and the
domination numbers of $\Gamma(R)$ and $\Gamma(R[x,\alpha,\delta])$,
where $R[x,\alpha,\delta]$ is the Ore extension of $R$, are studied.
Keywords:zerodivisor graph, domination number Categories:05C75, 13H10 

5. CMB 2013 (vol 57 pp. 188)
6. CMB 2011 (vol 56 pp. 407)
 Rad, Nader Jafari; Jafari, Sayyed Heidar; Mojdeh, Doost Ali

On Domination in ZeroDivisor Graphs
We first determine the domination number for the zerodivisor
graph of the product of two commutative rings with $1$. We then
calculate the domination number for the zerodivisor graph of any
commutative artinian ring. Finally, we extend some of the results
to noncommutative rings in which an element is a left
zerodivisor if and only if it is a right zerodivisor.
Keywords:zerodivisor graph, domination Categories:13AXX, 05C69 
