Volume 35 • Number 2 • 2012   • About a Conjecture on the Randi$\acute{c}$ Index of Graphs Liancui Zuo Abstract. For an edge $uv$ of a graph $G$, the weight of the edge $e=uv$ is defined by $w(e)=1/\sqrt{d(u)d(v)}$. Then $$R(G)=\sum_{uv\in E(G)}1/\sqrt{d(u)d(v)}=\sum_{e\in E(G)}w(e)$$ is called the Randi$\acute{c}$ index of $G$. If $G$ is a connected graph, then $${\rm rad}(G)=\min_{x}\max_{y} d(x,y)$$ is called the radius of $G$, where $d(x,y)$ is the distance between two vertices $x,y$. In $2000$, Caporossi and Hansen conjectured that for all connected graphs except the even paths, $R(G)\geq r(G)$. They proved the conjecture holds for all trees except the even paths. In this paper, it is proved that the conjecture holds for all unicyclic graphs, bicyclic graphs and some class of chemical graphs. 2010 Mathematics Subject Classification: 05C75, 05C90. Full text: PDF