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 E (G) ≤ (n / 2) (1 + sqrt(n)) for any graph G of order n and exhibited an infinite family of graphs with E (G) = (v (G) / 2) (1 + sqrt(v (G))). 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. © 2006 Elsevier Inc. All rights reserved.
Journal of Mathematical Analysis and Applications
Nikiforov, V. (2007). Graphs and matrices with maximal energy. Journal of Mathematical Analysis and Applications, 327 (1), 735-738. https://doi.org/10.1016/j.jmaa.2006.03.089