The energy of graphs and matrices
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. Let 2 ≤ m ≤ n, A be an m × n nonnegative matrix with maximum entry α, and {norm of matrix} A {norm of matrix}1 ≥ n α. Extending previous results of Koolen and Moulton for graphs, we prove that{Mathematical expression} Furthermore, if A is any nonconstant matrix, thenE (A) ≥ σ1 (A) + frac({norm of matrix} A {norm of matrix}22 - σ12 (A), σ2 (A)) . Finally, we note that Wigner's semicircle law implies thatE (G) = (frac(4, 3 π) + o (1)) n3 / 2 for almost all graphs G. © 2006 Elsevier Inc. All rights reserved.
Publication Title
Journal of Mathematical Analysis and Applications
Recommended Citation
Nikiforov, V. (2007). The energy of graphs and matrices. Journal of Mathematical Analysis and Applications, 326 (2), 1472-1475. https://doi.org/10.1016/j.jmaa.2006.03.072
