Sandwiching the (generalized) Randić index

Abstract

The well-known Randić index of a graph $G$ is defined as $R\left(G\right)=\sum {(d_u\cdot d_v)}^{-1/2}$, where the sum is taken over all edges $uv\in E\left(G\right)$ and $d_u$ and $d_v$ denote the degrees of $u$ and $v$, respectively. Recently, it was found useful to use its simplified modification: $R^{\prime }\left(G\right)=\sum {(max\{d_u,d_v\})}^{-1}$, which represents a lower bound for the Randić index. In this paper we introduce generalizations of $R^{\prime }$ and its counterpart, $R^{″}$, defined as $R_{\alpha }^{\prime }\left(G\right)=\sum min\{{d_u}^{\alpha },{d_v}^{\alpha }\}$ and $R_{\alpha }^{″}\left(G\right)=\sum max\{{d_u}^{\alpha },{d_v}^{\alpha }\}$, for any real number $\alpha$. Clearly, the former is a lower bound for the generalized Randić index, and the latter is its upper bound. We study extremal values of $R_{\alpha }^{\prime }$ and $R_{\alpha }^{″}$, and present extremal graphs within the classes of connected graphs and trees. We conclude the paper with several problems.