SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH

Abstract

Let$G$be a simple connected graph with$n$vertices and$m$edges and$d_{1}\geq d_{2}\geq \cdots \geq d_{n}>0$its sequence of vertex degrees. If$\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}>\unicode[STIX]{x1D707}_{n}=0$are the Laplacian eigenvalues of$G$, then the Kirchhoff index of$G$is$\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$. We prove some new lower bounds for$\mathit{Kf}(G)$in terms of some of the parameters$\unicode[STIX]{x1D6E5}=d_{1}$,$\unicode[STIX]{x1D6E5}_{2}=d_{2}$,$\unicode[STIX]{x1D6E5}_{3}=d_{3}$,$\unicode[STIX]{x1D6FF}=d_{n}$,$\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$and the topological index$\mathit{NK}=\prod _{i=1}^{n}d_{i}$.