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

Abstract It is known that the spectral radius of a digraph with k edges is ≤ √ k, and that this inequality is strict except when k is a perfect square. For k = m + l, l fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex and connecting it to the first l/2 vertices by pairs of directed edges. 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 s, k = (m + 1) − 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 s/2 pairs of directed edges that connect to the last vertex.