Asymptotic approximation for the number of graphs

We prove that, for fixed $k\geq3$, the following classes of labeled $n$-vertex graphs are asymptotically equicardinal: graphs of diameter $k$, connected graphs of diameter at least $k$, and (not necessarily connected) graphs with a shortest path of length at least $k$. An asymptotically exact approximation of the number of such $n$-vertex graphs is obtained, and an explicit error estimate in the approximation is found. Thus, the estimates are improved for the asymptotic approximation of the number of $n$-vertex graphs of fixed diameter $k$ earlier obtained by Füredi and Kim. It is shown that almost all graphs of diameter $k$ have a unique pair of diametrical vertices but almost all graphs of diameter 2 have more than one pair of such vertices. Illustr. 3, bibliogr. 9.