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

## TL;DR

This paper establishes sharp lower and upper bounds for the Randić index of graphs with given minimum and maximum degrees, characterizing the extremal graphs where equality holds.

## Contribution

It provides the first tight bounds for the Randić index based on degree constraints and characterizes the extremal graphs achieving these bounds.

## Key findings

- Lower bound for Randić index in terms of degrees and number of vertices.
- Upper bound for Randić index in connected graphs with degree constraints.
- Characterization of graphs where bounds are tight.

## 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 $R(G) \le \frac n2- \sum_{i=d}^{D-1}\frac 12 \left( \frac 1{\sqrt{i}} - \frac 1{\sqrt{i+1}}\right)^2$; it is sharp for infinitely many $n$, and we characterize when equality holds in the bound.