First and Second Maximum of Randi\'{c} Index Among all $k-$Cyclic Graphs of a Given Order

Abstract

Suppose $G$ is a simple graph with edge set $E(G)$. The Randi\'{c} index $R(G)$ is defined as $R(G)=\sum_{uv\in E(G)}\frac{1}{\sqrt{deg_{G}(u)deg_{G}(v)}}$, where $deg_G(u)$ denotes the vertex degree of $u$ in $G$. In this paper, the first and second maximum of Randi\'{c} index among all $n-$vertex $k-$cyclic graphs were computed.