A Lower bound for the spectral radius of a digraph

Linear Algebra and its Applications, 433(1) (2010), 233–240
http://dx.doi.org/10.1016/j.laa.2010.02.012

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.