Some properties of the essential annihilating-ideal graph of commutative rings

Let S be a commutative ring with unity and A(S) denotes the set of annihilating-ideals of S. The essential annihilating-ideal graph of S, denoted by EG(S), is an undirected graph with A∗(S) as the set of vertices and for distinct I, J ∈ A∗(S), I ∼ J is an edge if and only if Ann(IJ ) ≤e S. In this paper, we classify the Artinian rings S for which EG(S) is projective. We also discuss the coloring of EG(S). Moreover, we discuss the domination number of EG(S).