Graphs and matrices with maximal energy
Abstract
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.
Publication Title
Journal of Mathematical Analysis and Applications
Recommended Citation
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
