On a Relation Between Randic Index and Algebraic Connectivity

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 $P_n$,

关于连通图\(G\)的兰迪奇指数\(R\)和代数连通度\(a\)之间关系的一个AutoGraphiX猜想是： \(\frac{R}{a} \leq \frac{\frac{n - 3 + 2\sqrt{2}}{2}}{2(1 - \cos\frac{\pi}{n})}\) 当且仅当\(G\)是\(P_n\)（\(n\)阶路）时等号成立。