Asymptotic approximation for the number of n-vertex graphs of given diameter
We prove that, for fixed к > 3, the following classes of labeled n-vertex graphs are asymptotically equicardinal: graphs of diameter k, connected graphs of diameter at least к, and (not necessarily connected) graphs with a shortest path of length at least к. 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 к earlier obtained by Füredi and Kim. It is shown that almost all graphs of diameter к have a unique pair of diametrical vertices but almost all graphs of diameter 2 have more than one pair of such vertices.