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

Abstract

For a simple connected graph G of order n, having Laplacian eigenvalues μ_1, μ_2, . . . ,μ_{nâ1}, μ_n = 0, the Laplacianâenergyâlike invariant (LEL) and the Kirchhoff index (Kf) are defined as LEL(G) = \sum_{i=1}^{n-1} \sqrt{μ_i} Kf(G) = \sum_{i=1}^{n-1} 1/μ_i, respectively. In this paper, LEL and Kf arecompared, and sufficient conditions for the inequality Kf(G) < LEL(G) are established.