Hypoenergetic and Nonhypoenergetic Digraphs

Abstract 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 = 1 n σ i , where σ 1 ≥ ⋯ ≥ 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 σ 1 and N ( D ) in terms of the size of digraph D.