The maximum number of edges in a graph with fixed edge-degree

Suppose that n greater than or equal to 2t + 2 (t greater than or equal to 17). Let G be a graph with n vertices such that its complement is connected and, for all distinct non-adjacent vertices u and upsilon, there are at least t common neighbours. Then we prove that \E(G)\ greater than or equal to inverted right perpendicular(2t + 1)n - 2t(2) - 3)inverted left perpendicular /2 (n less than or equal to 3t - 1) and \E(G)\ greater than or equal to (t + 1)n - t(2) - t - 3 (n greater than or equal to 3t). Furthermore, the results are sharp.