%0 Journal Article
%T Some results on the complement of a new graph associated to a commutative ring
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Visweswaran, S.
%A Parmar, Anirudhdha
%D 2017
%\ 09/01/2017
%V 2
%N 2
%P 119-138
%! Some results on the complement of a new graph associated to a commutative ring
%K Annihilating ideal of a ring
%K maximal N-prime of (0)
%K connected graph
%K diameter. girth
%R 10.22049/cco.2017.25908.1053
%X The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. We denote the collection of all ideals of $R$ by $mathbb{I}(R)$ and $mathbb{I}(R)backslash {(0)}$ by $mathbb{I}(R)^{*}$. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, {em A new graph associated to a commutative ring}, Discrete Math. Algorithm. Appl. {bf 8} (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated to $R$, denoted by $Omega_{R}^{*}$ which is an undirected graph whose vertex set is $mathbb{I}(R)^{*}backslash {R}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if either $(Ann_{R}I)J = (0)$ or $(Ann_{R}J)I = (0)$. Several interesting theorems were proved on $Omega_{R}^{*}$ in the aforementioned paper and they illustrate the interplay between the graph-theoretic properties of $Omega_{R}^{*}$ and the ring-theoretic properties of $R$. The aim of this article is to investigate some properties of $(Omega_{R}^{*})^{c}$, the complement of the new graph $Omega_{R}^{*}$ associated to $R$.
%U http://comb-opt.azaruniv.ac.ir/article_13644_1b27eaa14546119e0ee5915425b1cb0b.pdf