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The weakly zero-divisor graph of a commutative ring
Volume 62, no. 1
(2021),
pp. 105–116
https://doi.org/10.33044/revuma.1677
Abstract
Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of
zero-divisors of $R$. The weakly zero-divisor graph of $R$ is the undirected
(simple) graph $W\Gamma(R)$ with vertex set $Z(R)^*$, and two distinct
vertices $x$ and $y$ are adjacent if and only if there exist $r\in \operatorname{ann}(x)$ and
$s\in \operatorname{ann}(y)$ such that $rs=0$. It follows that $W\Gamma(R)$ contains the
zero-divisor graph $\Gamma(R)$ as a subgraph. In this paper, the
connectedness, diameter, and girth of $W\Gamma(R)$ are investigated. Moreover,
we determine all rings whose weakly zero-divisor graphs are star.
We also give conditions under which weakly zero-divisor and
zero-divisor graphs are identical. Finally, the chromatic number of
$W\Gamma(R)$ is studied.
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Published by the Unión Matemática Argentina |
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