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The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G) ≤ n − 2 if G is a simple graph on n vertices and G is not
isomorphic to nK1. In this paper, we characterize the extremal graphs attaining the upper bound n − 2 and the second upper bound n − 3. The maximum nullity of simple graphs with n vertices and e edges, M(n, e), is also discussed. We obtain an upper bound of M(n, e), and characterize n and e for which the upper bound is achieved.