Abstract: For a commutative ring R, its essential graph EG(R) is an undirected simple graph whose vertex set is Z(R)\0, and two distinct vertices x and y are adjacent if and only if ann(xy) is an essential ideal. By giving a necessary and sufficient condition for Z_n such that its zero-divisor graph coincides with its essential graph, it is showed that a bipartite essential graph of a commutative ring must be a complete bipartite graph, and the classifications of the corresponding rings up to isomorphism are also established.