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# Some Results on the Annihilating-ideal Graphs

Published:2016-05-10
Printed: Sep 2016
Department of Mathematics, Faculty of Sciences, Razi University , Kermanshah, Iran
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## Abstract

The annihilating-ideal graph of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, 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 $IJ=(0)$. Here, we show that if $R$ is a reduced ring and the independence number of $\mathbb{AG}(R)$ is finite, then the edge chromatic number of $\mathbb{AG}(R)$ equals its maximum degree and this number equals $2^{|{\rm Min}(R)|-1}-1$; also, it is proved that the independence number of $\mathbb{AG}(R)$ equals $2^{|{\rm Min}(R)|-1}$, where ${\rm Min}(R)$ 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 $\mathbb{AG}(R)$ is not Eulerian, and it is Hamiltonian if and only if $R$ contains no Gorenstain ring as its direct summand.
 Keywords: annihilating-ideal graph, independence number, edge chromatic number, bipartite, cycle
 MSC Classifications: 05C15 - Coloring of graphs and hypergraphs 05C69 - Dominating sets, independent sets, cliques 13E05 - Noetherian rings and modules 13E10 - Artinian rings and modules, finite-dimensional algebras

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