On Kirchhoff and Degree Kirchhoff Indices

Igor Milovanović, Ivan Gutman, Emina Milovanović

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 $Kf(G)=n\sum_{i=1}^{n-1}1/\mu_i$ and $DKf(G)=2m\sum_{i=1}^{n-1}1/\rho_i$, respectively. Upper bounds for $Kf(G)$ and $DKf(G)$ are obtained.