The energy of graphs and matrices

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:326

Article

Nikiforov, Vladimir

关键词: graph energy;    graph eigenvalues;    singular values;    matrix energy;    Wigner's semicircle law;   

DOI  :  10.1016/j.jmaa.2006.03.072

来源: Elsevier

【 摘 要 】

Given a complex m x n matrix A, we index its singular values as sigma(1) (A) >= sigma(2) (A) >= (...) and call the value epsilon (A) = sigma(1) (A) + sigma(2) (A) + (...) the energy of A, thereby extending the concept of graph energy, introduced by Gutman. Let 2 <= m <= n, A be an m x n nonnegative matrix with maximum entry alpha, and parallel to A parallel to(1) >= n alpha. Extending previous results of Koolen and Moulton for graphs, we prove that epsilon(A) <= parallel to A parallel to(1)/root(m-1)(parallel to A parallel to(2)(1)/mn) <= alpha root n(m + root m)/2. Furthermore, if A is any nonconstant matrix, then epsilon(A) >= sigma(1) (A) + parallel to A parallel to(2)(2)-sigma(2)(1)(A)/sigma(2)(A). Finally, we note that Wigner's semicircle law implies that epsilon(G) = (4/3 pi + o(1))n(3/2) for almost all graphs G. (c) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

Free