Some results on the complement of a new graph associated to a commutative ring

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 I(R) and I(R)\{(0)} by I(R) ^∗. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, A new graph associated to a commutative ring, Discrete Math. Algorithm. Appl. 8 (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated to R, denoted by Ω^∗ R which is an undirected graph whose vertex set is I(R) ^∗\{R} and distinct vertices I, J are joined by an edge in this graph if and only if either (AnnRI)J = (0) or (AnnRJ)I = (0). Several interesting theorems were proved on Ω^∗ R in the aforementioned paper and they illustrate the interplay between the graph-theoretic properties of Ω^∗ R and the ring-theoretic properties of R. The aim of this article is to investigate some properties of (Ω^∗ R) c , the complement of the new graph Ω^∗ R associated to R.