  Let $n,k,t$ be positive integers. What is the maximum number of arcs in a
digraph on $n$ vertices in which there are at most $t$ distinct walks of length
$k$ with the same endpoints? In this paper, we prove that the maximum number is
equal to $n(n-1)/2$ and the extremal digraph are the transitive tournaments
when $k\ge n-1\ge \max\{2t+1,2\left\lceil \sqrt{2t+9/4}+1/2\right\rceil+3\}$.
Based on this result, we may determine the maximum numbers and the extremal
digraphs for $k\ge \max\{2t+1,2\left\lceil \sqrt{2t+9/4}+1/2\right\rceil+3\}$
and $n$ is sufficiently large, which generalises the existing results. A
conjecture is also presented.
