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

Abstract

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).