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 A 1 nα. Extending previous results of Koolen and Moulton for graphs, we prove that E(A) A 1 √mn + (m− 1) A 22 − A 21 mn α √n(m+√m) 2 . Furthermore, if A is any nonconstant matrix, then E(A) σ1(A)+ A 22 − σ2 1 (A) σ2(A) . Finally, we note that Wigner’s semicircle law implies that E(G) = 4 3π + o(1) n3/2 for almost all graphs G. © 2006 Elsevier Inc. All rights reserved