Upper bound for the energy of strongly connected digraphs

Abstract

The energy of a digraph D is defined as E(D) = ?n,i=1 ?Re(zi)?, 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 (n, n+?n/2, 3n+2?n/8, n+2?n/8, n+2?n/8). 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.