On Kirchhoff and Degree Kirchhoff Indices

Abstract

Let $G$ be an undirected connected graph with $n$ vertices and $m$ edges. If $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{n-1} > \mu_n =0$ and $\rho_1 \geq \rho_2 \geq \cdots \geq \rho_{n-1} > \rho_n =0$ are the Laplacian and the normalized Laplacian eigenvalues of $G$, then the Kirchhoff and the degree Kirchhoff indices obey the relations $Kf(G) = n\sum\limits_{i=1}^{n-1} 1/\mu_i$ and $DKf(G) = 2m \sum\limits_{i=1}^{n-1} 1/\rho_i$\,, respectively. Upper bounds for $Kf(G)$ and $DKf(G)$ are obtained.