Sharp bounds for the Randic index of graphs with given minimum and maximum degree

Abstract

The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac{1}{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)={\sum }_{uv\in E(G)}\frac{1}{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it. Let $d$ and $D$ be positive integers $d<D$. In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G)\ge \frac{\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex $(d,D)$-biregular. Furthermore, we show that if $G$ is an $n$-vertex connected graph with minimum degree $d$ and maximum degree $D$, then $R(G)\le \frac{n}{2}-{\sum }_{i=d}^{D-1}\frac{1}{2}{\left(\frac{1}{\sqrt{i}}-\frac{1}{\sqrt{i+1}}\right)}^2$; it is sharp for infinitely many $n$, and we characterize when equality holds in the bound.

7 pages