Randi\'c index, diameter and the average distance

TL;DR

This paper investigates the relationships between the Randić index, diameter, and average distance of graphs, proving several bounds and partial solutions to existing conjectures for graphs with certain minimum degree conditions.

Contribution

It provides new bounds linking the Randić index with diameter and average distance, partially resolving conjectures for graphs with specified minimum degrees.

Findings

Proves bounds on Randić index related to diameter for graphs with minimum degree ≥ 5.

Establishes inequalities involving Randić index, diameter, and average distance for large graphs.

Offers conditions under which the Randić index exceeds the average distance and ratio bounds hold.

Abstract

The Randi\'c index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randi\'c index $R(G)$ with relations to the diameter $D(G)$ and the average distance $\mu(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $\delta(G)$, if $\delta(G)\geq 5$, then $R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$; if $\delta(G)\geq n/5$ and $n\geq 15$, $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq \mu(G)$. Furthermore, for any arbitrary real number $\varepsilon \ (0<\varepsilon<1)$, if $\delta(G)\geq \varepsilon n$, then $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq \mu(G)$ hold for sufficiently large $n$.