Some relations between rank of a graph and its complement

Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n + 2. Also for every integer n ⩾ 5 and any k, 0 ⩽ k ⩽ n, we construct a graph G of order n, such that .