Ela the Maximal Spectral Radius of a Digraph With

It is known that the spectral radius of a digraph with k edges is at most √ k, and that this inequality is strict except when k is a perfect square. For k = m2 + , fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex, a corresponding loop, and then connecting it to the first /2 vertices by pairs of directed edges (for even an extra edge is added to the new vertex). Using a combinatorial reciprocity theorem, and a classification by Backelin on the digraphs on s edges having a maximal number of walks of length two, the following result is obtained: for fixed 0 < s = 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 s/2 pairs of directed edges that connect to the last vertex (if s is even, remove an extra edge connecting the last vertex).