Cao's strengthened Randić index–radius conjecture

Let $G$ be a simple undirected connected graph with vertex set $V(G)=\{v_1,\dots,v_n\}$ and edge set $E(G)$. Let $d_i$ be the degree of $v_i$, define the Randić index by

$$R(G)=\sum_{v_i v_j \in E(G)} \frac{1}{\sqrt{d_i d_j}},$$

and let $r(G)=\min\limits_{v_i\in V(G)}\max\limits_{v_j\in V(G)}\rho(v_i,v_j)$ be the graph radius, where $\rho(v_i,v_j)$ is the distance between two vertices.

Cao's strengthened conjecture. For any connected graph $G$ except even paths,

$$R(G)\geq r(G).$$

This is a stronger version of the Randić index–radius conjecture. The paper records the inequality for trees, apart from even paths, and leaves the assertion for general connected graphs open.