  In 1984, Plesn\'{i}k determined the minimum total distance for given order
and diameter and characterized the extremal graphs and digraphs. We prove the
analog for given order and radius, when the order is sufficiently large
compared to the radius. This confirms asymptotically a conjecture of Chen et
al. We show the connection between minimizing the total distance and maximizing
the size under the same conditions. We also prove some asymptotically optimal
bounds for the maximum total distance.
