The Randić index R(G) of a nontrivial connected graph G is defined as the sum of the weights (d(u)d(v))−12 over all edges e=uv 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+222n−2 conjectured by Aouchiche, Hansen and Zheng (2007) [1].