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

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