Research articles
Properties of annihilator graph of a commutative semigroup
Yahya Talebi,
Sahar Akbarzadeh Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran
Аннотация:
Let
$S$ be a commutative semigroup with zero. Let
$Z(S)$ be the set of all zero-divisors of
$S$. We define the annihilator graph of
$S$, denoted by
$ANN_{G}(S)$, as the undirected graph whose set of vertices is
$Z(S)^{\ast}=Z(S)-\{0\}$, and two distinct vertices
$x$ and
$y$ are adjacent if and only if
$ann_{S}(xy)\neq ann_{S}(x)\cap ann_{S}(y)$. In this paper, we study some basic properties of
$ANN_{G}(S)$ by means of
$\Gamma(S)$. We also show that if
$Z(S)\neq S$, then
$ANN_{G}(S)$ is a subgraph of
$\Gamma(S)$. Moreover, we investigate some properties of the annihilator graph
$ANN_{G}(S)$ related to minimal prime ideals of
$S$. We also study some connections between the domination numbers of annihilator graphs and zero-divisor graphs.
Ключевые слова и фразы:
Annihilator graph, diameter, girth, zero divisor graph.
MSC: 20M14,
05C75 Поступила в редакцию: 03.07.2017
Исправленный вариант: 23.07.2018
Язык публикации: английский