Bounds and power means for the general Randic index

TL;DR

This paper reviews bounds for the general Randić index, employing the power mean inequality to derive new bounds and generalizations, including spectral and non-spectral inequalities involving the graph's spectral radius.

Contribution

It introduces new bounds for the general Randić index using power mean inequalities and extends existing bounds to broader cases, including spectral bounds.

Findings

Established lower bounds for R_eta using spectral radius.

Generalized non-spectral bounds for the zeroth-order Randić index.

Strengthened existing bounds for R_eta with new inequalities.

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 $\alpha < 0$.