Upper bound for the energy of strongly connected digraphs

Abstract

The energy of a digraph D is dened as E(D), where z1, z2,..., zn are the (possibly complex) eigenvalues of D. We show that if D is a strongly connected digraph on n vertices, a arcs, and c2 closed walks of length two, such that Re(z1) ≥ (a + c2)=(2n) ≥ 1, then E(D) ≤ n(1 + √n)/2. Equality holds if and only if D is a directed strongly regular graph with parameters. This bound extends to digraphs an earlier result [J. H. Koolen, V. Moulton: Maximal energy graphs. Adv. Appl. Math., 26 (2001), 47-52], obtained for simple graphs.