Tur\'an problems for digraphs avoiding distinct walks of a given length   with the same endpoints
  Let $n \ge 5$ and $k\ge 4$ be positive integers. We determine the maximum size of digraphs of order n that avoid distinct walks of length k with the same endpoints. We also characterize the extremal digraphs attaining this maximum number when $k \ge 5$. 
Zejun Huang, Zhenhua Lyu, Pu Qiao
math.CO