The average distance and the diameter of dense random regular graphs

Let AD(G(n,d)) be the average distance of G(n,d), a random n-vertex d-regular graph. For d = (beta + o(1))n(alpha) with two arbitrary constants alpha is an element of (0,1) and beta > 0, we prove that vertical bar AD(G(n,d)) - j mu vertical bar < epsilon holds with high probability for any constant epsilon > 0, where mu is equal to alpha(-1) + exp(-beta(1/alpha)) if alpha(-1) is an element of N and to [alpha(-1)] otherwise. Consequently, we show that the diameter of the G(n,d) is equal to left perndicular alpha(-1) right perpendicular +1 with high probability.