On the Randić index and girth of graphs

Abstract

The Randić index $R\left(G\right)$ of a graph $G$ is defined by $R\left(G\right)={\sum }_{uv}\frac{1}{\sqrt{d\left(u\right)d\left(v\right)}}$, where $d\left(u\right)$ is the degree of a vertex $u$ in $G$ and the summation extends over all edges $uv$ of $G$. In this work, we give a sharp upper bound and a lower bound of the Randić index among connected $n$-vertex graphs with girth $g\ge k(k\ge 3)$.