Comparison between Kirchhoff index and the Laplacian-energy-like invariant

Abstract

Let G be a connected graph of order n with Laplacian eigenvalues μ1≥ μ2≥⋯≥μn- 1> μn=0. The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as Kf=n∑k=1n-11/ μk and LEL=∑k=1n-1 μk, respectively. We compare Kf and LEL and establish two sufficient conditions under which LEL<Kf. The connected graphs of order n with nine greatest Kirchhoff indices are determined; for these LEL>Kf holds. © 2011 Elsevier Inc. All rights reserved.