On the Tree with Maximum General Randic Index

Abstract

The general Randic index $R_{-\alpha }(G)$ of a graph $G$, defined by a real number $\alpha$, is the sum of $(d(u)d(v){)}^{-\alpha }$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we have discussed some properties of the Max Tree which has the maximum general Randic index $R_{-\alpha }(G)$, where $\alpha \in ({\alpha }_0,2)$. Based on these properties, we are able to obtain the structure of the Max Tree among all trees of order $k\ge 3$. Thus, the maximal value of $R_{-\alpha }(G)$ follows easily.