The rings considered in this paper are commutative with identity that are not integral domains. Let R be a ring. Let \(\mathbb {A}(R)\) denote the set of all annihilating ideals of R, and let us denote \(\mathbb {A}(R)\backslash \{(0)\}\) by \(\mathbb {A}(R)^{*}\) . For any ideal I of R, we denote the annihilator of I in R by \(Ann_{R}(I)\) . Recall that the weakly annihilating-ideal graph of R, denoted by \(\mathbb {WAG}(R)\) , is an undirected graph whose vertex set is \(\mathbb {A}(R)^{*}\) and distinct vertices I and J are adjacent if and only if there exist nonzero ideals A and B of R with \(A\subseteq Ann_{R}(I)\) and \(B\subseteq Ann_{R}(J)\) such that \(AB = (0)\) . We denote the complement of \(\mathbb {WAG}(R)\) by \((\mathbb {WAG}(R))^{c}\) . With the assumption that R is reduced, this paper aims to study the interplay between the graph-theoretic properties of \((\mathbb {WAG}(R))^{c}\) and the ring-theoretic properties of R.
