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The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \inE(G)} \frac 1{\sqrt{d(u)d(v)}}$, 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 $R(G) \ge\frac{\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex$(d,D)$-biregular. Furthermore, we show that if $G$ is an $n$-vertex connectedgraph with minimum degree $d$ and maximum degree $D$, then $R(G) \le \frac n2-\sum_{i=d}^{D-1}\frac 12 \left( \frac 1{\sqrt{i}} - \frac1{\sqrt{i+1}}\right)^2$; it is sharp for infinitely many $n$, and wecharacterize when equality holds in the bound.

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