The structure of the typical graphs of given diameter

In this paper it is proved that there are constants 0 < c 2 < c 1 such that the number of (labeled) n -vertex graphs of diameter d is ( 1 + o ( 1 ) ) d − 2 2 n ( d − 1 ) 3 n − d + 1 2 n − d + 1 2 whenever n → ∞ and 3 ≤ d ≤ n − c 1 log n , where n ( d − 1 ) = n ( n − 1 ) … ( n − d + 2 ) . A typical graph of diameter d consists of a combination of an induced path of length d and a highly connected block of size n − d + 3 . In the case d > n − c 2 log n the typical graph has a completely different snakelike structure. The number of n -vertex graphs of diameter d is ( 1 + o ( 1 ) ) 1 2 n ( d + 1 ) 3 n − d − 1 d n − d − 1 whenever n → ∞ and d > n − c 2 log n .