Lower bounds for the Randić index in terms of matching number

Saieed Akbari, Sina Ghasemi Nezhad, Reyhane Ghazizadeh, John Haslegrave, Elahe Tohidi

Published 2024-02-20, updated 2024-08-20Version 2

We investigate how small the Randi\'c index of a graph can be in terms of its matching number, and prove several results. We give best-possible linear bounds for graphs of small excess and for subcubic graphs; in the former case the size of excess we permit is qualitatively the best possible. We show that a linear bound holds for any sparse hereditary graph class (such as planar graphs). In general, however, we show that it can be much smaller than linear. We determine the asymptotic growth rate of the minimum Randi\'c index for graphs with a near perfect matching, and conjecture that the same bounds hold for all graphs.