A $k^{2/3}$ lower bound for the Randić index in terms of matching number

Let $G$ be a graph, let $R(G)$ denote its Randić index, and let $\alpha'(G)$ denote its matching number.

Randić-index matching conjecture. If

$$\alpha'(G)=k,$$

then

$$R(G)\geq ck^{2/3}$$

for some absolute constant $c>0$.

The paper proves the corresponding asymptotic lower bound for graphs having a nearly-perfect matching and conjectures that the same order of growth holds for all graphs.