  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.
