  It is known that the spectral radius of a digraph with k edges is \le
\sqrt{k}, and that this inequality is strict except when k is a perfect square.
For k=m^2 + \ell, \ell fixed, m large, Friedland showed that the optimal
digraph is obtained from the complete digraph on m vertices by adding one extra
vertex, and a corresponding loop, and then connecting it to the first \lfloor
\ell/2\rfloor vertices by pairs of directed edges (this is for odd \ell, for
even \ell we add one extra edge to the new vertex).
  Using a combinatorial reciprocity theorem by Gessel, and a classification by
Backelin on the digraphs on s edges having a maximal number of walks of length
two, we obtain the following result: for fixed 0< s \neq 4, k=(m+1)^2 - s, m
large, the maximal spectral radius of a digraph with k edges is obtained by the
digraph which is constructed from the complete digraph on m+1 vertices by
removing the loop at the last vertex together with \lfloor s/2 \rfloor pairs of
directed edges that connect to the last vertex (if s is even, remove an extra
edge connecting to the last vertex).
