Extremal values for the variation of the Randić index of bicyclic graphs

Abstract

Let $G$ be a connected graph of order $n$. The variation of the Randić index of $G$ is defined as

$$R^{\prime }\left(G\right)=\sum _{uv\in E\left(G\right)}\frac{1}{max\left\{d\left(u\right),d\left(v\right)\right\}},$$

where the summation goes over all edges $uv$ of $G$ and $d\left(u\right)$ is the degree of the vertex $u$ in $G$. In this paper, among all bicyclic graphs of order $n$, the minimum and maximum values for $R^{\prime }$ are determined, respectively.