The essential annihilating-ideal graph $$\mathcal{E}\mathcal{G}(R)$$ E G ( R ) of a commutative unital ring R is a simple graph, whose vertices are non-zero ideals of R with non-zero annihilator and there exists an edge between two distinct vertices I , J if and only if Ann ( IJ ) has a non-zero intersection with any non-zero ideal of R . In this paper, we show that $$\mathcal{E}\mathcal{G}(R)$$ E G ( R ) is weakly perfect, if R is Noetherian and an explicit formula for the clique number of $$\mathcal{E}\mathcal{G}(R)$$ E G ( R ) is given. Moreover, the structures of all rings whose essential annihilating-ideal graphs have chromatic number 2 are fully determined. Among other results, twin-free clique number and edge chromatic number of $$\mathcal{E}\mathcal{G}(R)$$ E G ( R ) are examined.