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

Abstract

Let G be a connected graph of order n with Laplacian eigenvalues ${\mu }_1⩾{\mu }_2⩾\cdots ⩾{\mu }_{n-1}>{\mu }_n=0$. The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as $\mathit{Kf}=n{\sum }_{k=1}^{n-1}1/{\mu }_k$ and $\mathit{LEL}={\sum }_{k=1}^{n-1}\sqrt{{\mu }_k}$, respectively. We compare Kf and LEL and establish two sufficient conditions under which $\mathit{LEL}<\mathit{Kf}$. The connected graphs of order n with nine greatest Kirchhoff indices are determined; for these $\mathit{LEL}>\mathit{Kf}$ holds.