1. CMB Online first
||Zero-divisor graphs of Ore extensions over reversible rings|
Let $R$ be an associative ring with identity.
First we prove some results about zero-divisor graphs of reversible
rings. Then we study the zero-divisors 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 zero-divisor
graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$,
$R$ is reversible and $(\alpha,\delta)$-compatible.
Keywords:zero-divisor graphs, reversible rings, McCoy rings, polynomial rings, power series rings
Categories:13B25, 05C12, 16S36
2. CMB 2015 (vol 58 pp. 271)
3. CMB 2014 (vol 57 pp. 573)
||Some Results on the Domination Number of a Zero-divisor Graph|
In this paper, we investigate the domination, total domination and
semi-total domination numbers of a zero-divisor 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:zero-divisor graph, domination number
4. CMB 2013 (vol 57 pp. 188)
||A Characterization of Bipartite Zero-divisor Graphs|
In this paper we obtain a characterization for all bipartite
zero-divisor graphs of commutative rings $R$ with $1$, such that
$R$ is finite or $|Nil(R)|\neq2$.
Keywords:zero-divisor graph, bipartite graph
5. CMB 2011 (vol 56 pp. 407)
||On Domination in Zero-Divisor Graphs|
We first determine the domination number for the zero-divisor
graph of the product of two commutative rings with $1$. We then
calculate the domination number for the zero-divisor graph of any
commutative artinian ring. Finally, we extend some of the results
to non-commutative rings in which an element is a left
zero-divisor if and only if it is a right zero-divisor.
Keywords:zero-divisor graph, domination