@article {
author = {Visweswaran, S. and Parmar, Anirudhdha},
title = {Some results on the complement of a new graph associated to a commutative ring},
journal = {Communications in Combinatorics and Optimization},
volume = {2},
number = {2},
pages = {119-138},
year = {2017},
publisher = {Azarbaijan Shahid Madani University},
issn = {2538-2128},
eissn = {2538-2136},
doi = {10.22049/cco.2017.25908.1053},
abstract = {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 $R$.},
keywords = {Annihilating ideal of a ring,maximal N-prime of (0),connected graph,diameter. girth},
url = {http://comb-opt.azaruniv.ac.ir/article_13644.html},
eprint = {http://comb-opt.azaruniv.ac.ir/article_13644_1b27eaa14546119e0ee5915425b1cb0b.pdf}
}