Digraphs that have at most one walk of a given length with the same endpoints

Let Θ(n,k)Θ(n,k) be the set of digraphs of order nn that have at most one walk of length kk with the same endpoints. Let θ(n,k)θ(n,k) be the maximum number of arcs of a digraph in Θ(n,k)Θ(n,k). We prove that if n≥5n≥5 and k≥n−1k≥n−1 then θ(n,k)=n(n−1)/2θ(n,k)=n(n−1)/2 and this maximum number is attained at DD if and only if DD is a transitive tournament. θ(n,n−2)θ(n,n−2) and θ(n,n−3)θ(n,n−3) are also determined.