We show that the spectral radius ?(D) of a digraph D with n vertices and c2 closed walks of length 2 satisfies ?(D)?c2/n. Moreover, equality occurs if and only if D is the symmetric digraph associated to a c2/n-regular graph, plus some arcs that do not belong to cycles. As an application of this result, we construct new sharp upper bounds for the low energy of a digraph, which extends Koolen and Moulton bounds of the energy to digraphs.

CEMAT - Center for Computational and Stochastic Mathematics