Sharper Randić index–diameter ratio conjecture for cactus graphs

Let $G$ be a cactus with $k$ cycles and $b$ bridges; write $n$ for its number of vertices, $R$ for its Randić index, and $d$ for its diameter. A BC-tree is the block-cut tree of $G$, and a graph is longitudinally symmetric when it has the symmetry specified by the paper. Cactus diameter-ratio conjecture. For a nontrivial cactus ($b=0$),

$$\frac{R}{d}\geq \frac{n-(k-1)(3-2\sqrt 2)}{n+k-1},$$

with equality if the BC-tree is a path and is longitudinally symmetric. For a cactus with $k>0$ and $b>0$,

$$\frac{R}{d}\geq \frac{n-(k-1)(3-2\sqrt 2)-6+\frac{4}{\sqrt 3}+4\sqrt{\frac{2}{3}}}{n+k+b-1},$$

with equality if the BC-tree is a path, the graph has two leaves, and it is longitudinally symmetric. This is proposed as a sharper diameter-ratio bound incorporating cycles and bridges; the supplied text says it is proved only for cacti whose BC-tree is starlike.