The Randić index and the diameter of graphs

The Randić index  R(G)R(G) of a graph GG is defined as the sum of 1dudv over all edges uvuv of GG, where dudu and dvdv are the degrees of vertices uu and vv, respectively. Let D(G)D(G) be the diameter of GG when GG is connected. Aouchiche et al. (2007) [1] conjectured that among all connected graphs GG on nn vertices the path PnPn achieves the minimum values for both R(G)/D(G)R(G)/D(G) and R(G)−D(G)R(G)−D(G). We prove this conjecture completely. In fact, we prove a stronger theorem: If GG is a connected graph, then R(G)−12D(G)≥2−1, with equality if and only if GG is a path with at least three vertices.