Randić index and the diameter of a graph

The Randić index R ( G ) of a nontrivial connected graph G is defined as the sum of the weights ( d ( u ) d ( v ) ) − 1 2 over all edges e = u v of G . We prove that R ( G ) ≥ d ( G ) / 2 , where d ( G ) is the diameter of G . This immediately implies that R ( G ) ≥ r ( G ) / 2 , which is the closest result to the well-known Graffiti conjecture R ( G ) ≥ r ( G ) − 1 of Fajtlowicz (1988) [4], where r ( G ) is the radius of G . Asymptotically, our result approaches the bound R ( G ) d ( G ) ≥ n − 3 + 2 2 2 n − 2 conjectured by Aouchiche, Hansen and Zheng (2007) [1].