Some Results on the Annihilating-ideal Graphs

Canadian mathematical bulletin

Farzad Shaveisi¹ 

[1] Department of Mathematics, Faculty of Sciences, Razi University , Kermanshah, Iran

关键词: annihilating-ideal graph;    independence number;    edge chromatic number;    bipartite;    cycle;   

DOI  :  10.4153/CMB-2016-016-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

The annihilating-ideal graphof a commutative ring $R$, denoted by $mathbb{AG}(R)$, is a graph whose vertex set consists of all non-zero annihilatingideals and two distinctvertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here,we show that if $R$ is a reduced ring and the independencenumber of $mathbb{AG}(R)$ is finite, then the edge chromaticnumber of $mathbb{AG}(R)$ equals its maximum degreeand this number equals $2^{|{m Min}(R)|-1}-1$; also, it is proved that the independence number of $mathbb{AG}(R)$ equals $2^{|{m Min}(R)|-1}$, where ${m Min}(R)$ denotes the setof minimal prime ideals of $R$.Then we give some criteria for a graph to be isomorphic withan 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 $R$ containsno Gorenstain ring as its direct summand.

【 授权许可】

Unknown