Lower bounds for the energy of digraphs

The energy of a digraph D is defined as , where z₁,…,z_n are the eigenvalues of D. In this article we find lower bounds for the energy of digraphs in terms of the number of closed walks of length 2, extending in this way the result obtained by Caporossi et al. G. Caporossi, D. Cvetkovi?, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996]: for all graphs G with m edges. Also, we study digraphs with three eigenvalues.