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Let AD(Gn,d) be the average distance of Gn,d, a random n-vertex d-regular graph. For d = (β + o(1))nα with two arbitrary constants α ∈ (0, 1) and β > 0, we prove that |AD(Gn,d) − μ| < holds with high probability for any constant > 0, where μ is equal to α−1 + exp(−β1/α) if α−1 ∈ N and to dα−1e otherwise. Consequently, we show that the diameter of the Gn,d is equal to bα−1c+1 with high probability. Mathematics Subject Classifications: 05C80, 05C12

The Average Distance and the Diameter of Dense Random Regular Graphs