The essential annihilating-ideal graph EG(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 EG(R) is weakly perfect, if R is Noetherian and an explicit formula for the clique number of EG(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 EG(R) are examined.