The energy of graphs and matrices


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