Randić index and the diameter of a graph

@article{Dvorak2011RandiIA,
  title={Randi{\'c} index and the diameter of a graph},
  author={Zdenek Dvorak and Bernard Lidick{\'y} and Riste Skrekovski},
  journal={Eur. J. Comb.},
  year={2011},
  volume={32},
  pages={434-442}
}
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 = uv ofG. 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 Grafiti conjecture R(G) ≥ r(G) − 1 of Fajtlowicz [4], where r(G) is the radius of G. Asymptotically, our result approaches the bound R(G) d(G) ≥ n−3+2 √ 2 2n−2 conjectured by Aouchiche, Hansen and Zheng [1]. 

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