Yongtang Shi

The Randi\'{c} index $R(G)$ of a graph $G$ is defined as the sum of the weights $(d(u)d(v))^{-\frac{1}{2}}$ of all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. B. Bollob\'as and P. Erd\"{o}s proved that the Randi\'{c} index of a graph with order $n$ without isolated vertices is at least $\sqrt{n-1}$. They asked for the minimum value of $R(G)$ for graphs $G$ with given minimum degree $\delta(G)$. C. Delorme et al answered their question of $\delta(G)=2$ and proposed a conjecture. In this paper, we verify the conjecture for $\delta(G)=3$. Furthermore, we prove that for a given minimal degree $\delta(G)\geq k\geq 4$, the conjecture is true when $n\geq \frac{3}{2}k^3$.