Note on a relation between Randic index and algebraic connectivity

TL;DR

This paper investigates a conjectured relationship between the Randić index and algebraic connectivity in graphs, proving it for various classes and establishing bounds for their product.

Contribution

The authors prove the conjecture for all trees, graphs with edge connectivity at least two, and graphs with certain diameter or degree conditions, and establish bounds on the product of Randić index and algebraic connectivity.

Findings

Conjecture holds for all trees.

Conjecture holds for graphs with edge connectivity ≥ 2.

Bounds on Randić index times algebraic connectivity are established.

Abstract

A conjecture of AutoGraphiX on the relation between the Randi\'c index $R$ and the algebraic connectivity $a$ of a connected graph $G$ is: $$\frac R a\leq (\frac{n-3+2\sqrt{2}}{2})/(2(1- \cos {\frac{\pi}{n}})) $$ with equality if and only if $G$ is $P_n$, which was proposed by Aouchiche and Hansen [M. Aouchiche and P. Hansen, A survey of automated conjectures in spectral graph theory, {\it Linear Algebra Appl.} {\bf 432}(2010), 2293--2322]. We prove that the conjecture holds for all trees and all connected graphs with edge connectivity $\kappa'(G)\geq 2$, and if $\kappa'(G)=1$, the conjecture holds for sufficiently large $n$. The conjecture also holds for all connected graphs with diameter $D\leq \frac {2(n-3+2\sqrt{2})}{\pi^2}$ or minimum degree $\delta\geq \frac n 2$. We also prove $R\cdot a\geq \frac {8\sqrt{n-1}}{nD^2}$ and $R\cdot a\geq \frac {n\delta(2\delta-n+2)} {2(n-1)}$, and then $R\cdot a$ is minimum for the path if $D\leq (n-1)^{1/4}$ or $\delta\geq \frac n 2-1$.