Extremal Values of Randić Index among Some Classes of Graphs

Abstract

Suppose $G$ is a simple graph with edge set $E\left(G\right)$ . The Randić index $R\left(G\right)$ is defined as $R\left(G\right)={\displaystyle {\sum }_{uv\in E\left(G\right)}\left(1/\sqrt{{\mathrm{deg}}_G\left(u\right){\mathrm{deg}}_G\left(v\right)}\right)}$ , where ${\mathrm{deg}}_G\left(u\right)$ and ${\mathrm{deg}}_G\left(v\right)$ denote the vertex degrees of $u$ and $v$ in $G$ , respectively. In this paper, the first and second maximum of Randić index among all $n-$ vertex $c-$ cyclic graphs was computed. As a consequence, it is proved that the Randić index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the Randić index of connected chemical graphs.