The extremal generalised Randi\'c index for a given degree range

Abstract

O and Shi proved that the Randi\'c index of any graph $G$ with minimum degree at least $\delta$ and maximum degree at most $\Delta$ is at least $\frac{\sqrt{\delta\Delta}}{\delta+\Delta}|G|$, with equality if and only if the graph is $(\delta,\Delta)$-biregular. In this note we give a short proof via a more general statement. We apply the latter to classify the graphs in any given degree range which minimise (or maximise) the generalised Randi\'c index for any exponent, and describe the transitions between different types of behaviour precisely.