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Some Results on Annihilating-ideal Graphs
Published online by Cambridge University Press: 20 November 2018
Abstract
The annihilating-ideal graph of a commutative ring $R$, denoted by
$\mathbb{A}\mathbb{G}\left( R \right)$, is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices
$I$ and
$J$ are adjacent if and only if
$IJ\,=\,\left( 0 \right)$. Here we show that if
$R$ is a reduced ring and the independence number of
$\mathbb{A}\mathbb{G}\left( R \right)$ is finite, then the edge chromatic number of
$\mathbb{A}\mathbb{G}\left( R \right)$ equals its maximum degree and this number equals
${{2}^{\left| \min \left( R \right) \right|-1}}-\,1$; also, it is proved that the independence number of
$\mathbb{A}\mathbb{G}\left( R \right)$ equals
${{2}^{\left| \min \left( R \right) \right|-1}}$, where
$\min \left( R \right)$ denotes the set of minimal prime ideals of
$R$. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a finite graph
$\mathbb{A}\mathbb{G}\left( R \right)$ is not Eulerian, and that it is Hamiltonian if and only if
$R$ contains no Gorenstain ring as its direct summand.
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- Research Article
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- Copyright © Canadian Mathematical Society 2016
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