Note on a relation between Randic index and algebraic connectivity

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 $\delta\geq \frac n 2-1$.