Some Results on Annihilating-ideal Graphs

Shaveisi, Farzad

Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 641-651

Résumé

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.

Détail

DOI : 10.4153/CMB-2016-016-3

Mots-clés : 05C15, 05C69, 13E05, 13E10, annihilating-ideal graph, independence number, edge chromatic number, bipartite, cycle