Randi\'c index, diameter and the average distance

The Randi'c index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1\mathrm{/}\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randi'c index $R(G)$ with relations to the diameter $D(G)$ and the average distance $\mu (G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $\delta (G)$, if $\delta (G)\ge 5$, then $R(G)-D(G)\ge \sqrt{2}-\frac{n+1}{2}$; if $\delta(G)\geq n/5$and$n\geq 15$,$\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$and$R(G)\geq \mu(G)$\mathrm{.}Furthermore,foranyarbitraryrealnumber$\varepsilon
(0<\varepsilon<1)$,if$\delta(G)\geq \varepsilon n$,then$\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$and$R(G)\geq \mu(G)$ hold for sufficiently large $n$.