Hypoenergetic and nonhypoenergetic digraphs

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑_i=1ⁿσ_i, where σ₁≥⋯≥σ_n are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N(D). For any digraph D it is proved that N(D)≥rank(D) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ₁ and N(D) in terms of the size of digraph D.