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

Suppose that n ⩾ 2t + 2 (t ⩾ 17). Let G be a graph with n vertices such that its complement is connected and, for all distinct non-adjacent vertices u and v, there are at least t common neighbours. Then we prove that |E(G)|≥⌈(2t+1)n−2t2-3)2 (n≤3t−1) and |E(G)z.sfnc;≥(t+1)n−t2−t−3 (n≥3t). Furthermore, the results are sharp.