Bounds and power means for the general Randic index

We review bounds for the general Randić index, $R_α = \sum_{ij \in E} (d_i d_j)^α$, and use the power mean inequality to prove, for example, that $R_α\ge mλ^{2α}$ for $α< 0$, where $λ$ is the spectral radius of a graph. This enables us to strengthen various known lower and upper bounds for $R_α$ and to generalise a non-spectral bound due to Bollobás \emph{et al}. We also prove that the zeroth-order general Randić index, $Q_α= \sum_{i \in V} d_i^α\ge nλ^α$ for $α< 0$.

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