Graphs and matrices with maximal energy

Given a complex m×n matrix A, we index its singular values as σ1(A)⩾σ2(A)⩾⋯ and call the value E(A)=σ1(A)+σ2(A)+⋯ the energy of A, thereby extending the concept of graph energy, introduced by Gutman. Koolen and Moulton proved that for any graph G of order n and exhibited an infinite family of graphs with . We prove that for all sufficiently large n, there exists a graph G=G(n) with E(G)⩾n3/2/2−n11/10. This implies a conjecture of Koolen and Moulton. We also characterize all square nonnegative matrices and all graphs with energy close to the maximal one. In particular, such graphs are quasi-random.