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Let R be a commutative ring with nonzero identity and let M be a unitary R-module. The essential graph of M , denoted by EG(M ) is a simple undirected graph whose vertex set is Z(M )\Ann R (M ) and two distinct vertices x and y are adjacent if and only if Ann M (xy) is an essential submodule of M . Let r(Ann R (M )) = Ann R (M ). It is shown that EG(M ) is a connected graph with diam(EG(M )) ≤ 2. Whenever M is Noetherian, it is shown that EG(M ) is a complete graph if and only if either Z(M ) = r(Ann R (M )) or EG(M ) = K 2 and diam(EG(M )) = 2 if and only if there are x, y ∈ Z(M )\Ann R (M ) and p ∈ Ass R (M ) such that xy ∈ p. Moreover, it is proved that gr(EG(M )) ∈ {3, ∞}. Furthermore, for a Noetherian module M with r(Ann R (M )) = Ann R (M ) it is proved that |Ass R (M )| = 2 if and only if EG(M ) is a complete bipartite graph that is not a star.

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