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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 optimaldigraph is obtained from the complete digraph on m vertices by adding one extravertex, 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, foreven \ell we add one extra edge to the new vertex).  Using a combinatorial reciprocity theorem by Gessel, and a classification byBackelin on the digraphs on s edges having a maximal number of walks of lengthtwo, we obtain the following result: for fixed 0< s \neq 4, k=(m+1)^2 - s, mlarge, the maximal spectral radius of a digraph with k edges is obtained by thedigraph which is constructed from the complete digraph on m+1 vertices byremoving the loop at the last vertex together with \lfloor s/2 \rfloor pairs ofdirected edges that connect to the last vertex (if s is even, remove an extraedge connecting to the last vertex).

The maximal spectral radius of a digraph with (m+1)^2-s edges