Bounds and power means for the general Randic index

Abstract

We review bounds for the general Randi\'c index, $R_{\alpha} = \sum_{ij \in E} (d_i d_j)^\alpha$, and use the power mean inequality to prove, for example, that $R_\alpha \ge m\lambda^{2\alpha}$ for $\alpha < 0$, where $\lambda$ is the spectral radius of a graph. This enables us to strengthen various known lower and upper bounds for $R_\alpha$ and to generalise a non-spectral bound due to Bollob\'as \emph{et al}. We also prove that the zeroth-order general Randi\'c index, $Q_\alpha = \sum_{i \in V} d_i^\alpha \ge n\lambda^\alpha$ for $\alpha < 0$.