Randic index and the diameter of a graph

The Randic 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 = 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+2 root 2/2n-2 conjectured by Aouchiche, Hansen and Zheng (2007) [1]. (C) 2010 Elsevier Ltd. All rights reserved.