Sharp bounds for the general Randić index $R_-1$ of a graph

Abstract

Let $G$ be an undirected simple, connected graph with $n \geq 3$ vertices and $m$ edges, with vertex degree sequence $d_1\ge d_2 \ge \cdots \ge d_n$. The general Randi\'c index is defined by \[ R_{-1}=\sum _{(i,j)\in E}\frac {1}{d_id_j}. \] Lower and upper bounds for $R_{-1}$ are obtained in this paper.