On the number of h-connected graphs with a fixed diameter

In this paper it is shown that in the class of labeled graphs of order n the proportion of h-connected graphs (h⩾2) of diameter equal to k, is equal to: (i) (0.75+o(1))n for k=3; (ii) (2−h−2+2−2+o(1))n for k=4, and (iii) ((2h+1−1)2−kh+3h−2+o(1))n for every k⩾5. This extends a result by the author (Discrete Math. 156 (1996) 219–228) on the number of graphs of order n and diameter equal to k (the case h=1).