The Randić Index and Girth of Triangle-Free Graphs

Abstract

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