Fajtlowicz's Randić index–radius conjecture

Let $G$) be a graph. Write $R=R(G)$ for its Randić index and $r$ for its radius. An even path is a path with an even number of vertices. Fajtlowicz's conjecture. If $G$ is an even path, then

$$R-r\geq \sqrt 2-\frac{3}{2};$$

otherwise,

$$R-r\geq 0.$$

The conjecture proposes the radius as a lower bound for the Randić index. The paper notes that the original conjecture was $R\geq r-1$, later strengthened to $R\geq r$ outside the even-path exception, and reports partial results before proving the bound for cactus graphs. Resolution in the full class of graphs is not established in the supplied text.