On a Conjecture Between RandiĆ Index and Average Distance of Unicyclic Graphs

Zhifu You, Bolian Liu

The Randi\'c index $R(G)$ of a graph $G$ is defined as $R(G)=\sum\limits_{uv\in E}{(d(u)d(v))^{-\frac12}}$, where the summation goes over all edges of $G$. In 1988, Fajtlowicz proposed a conjecture: For all connected graphs $G$ with average distance $ad(G)$, then $R(G)\geq ad(G)$. In this paper, we prove that this conjecture is true for unicyclic graphs.