Sharp bounds for the Randic index of graphs with given minimum and maximum degree

The Randi{' c} index of a graph $G$, written $R(G)$, is the sum of $\frac 1{\sqrt{d(u)d(v)}}$overalledges$uv$in$E(G)$\mathrm{.}$R(G)=\sum_{uv \in E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{' c} index of it. Let $d$ and $D$ be positive integers $d<D$. In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \ge \frac{\sqrt{dD}}{d+D}n$;equalityholdsonlywhen$G$isan$n$-vertex$(d,D)$-biregular\mathrm{.}Furthermore,weshowthatif$G$isan$n$-vertex connected graph with minimum degree $d$ and maximum degree $D$, then $R(G) \le \frac n2- \sum_{i=d}^{D-1}\frac 12 \left( \frac 1{\sqrt{i}} - \frac 1{\sqrt{i+1}}\right)^2$;itissharpforinfinitelymany$n$, and we characterize when equality holds in the bound.