Notes on "A proof for a conjecture on the Randić index of graphs with diameter"

Aouchiche et al. proposed a conjecture on the relationship between the Randic index R(G) and the diameter D(G) of a graph G: R(G)− D(G) ≥ √ 2 − n+1 2 and R(G) D(G) ≥ n−3+2 √ 2 2n−2 , with equalities if and only if G is a path. In [Jianxi Liu, Meili Liang, Bo Cheng, Bolian Liu, A proof for a conjecture on the Randic index of graphs with diameter. Appl. Math. Lett. 24(2011), 752− 756], the authors claimed that they proved the first part of the conjecture. But we find that their proof is invalid. This paper is to point out that the proofs of their two crucial lemmas are incorrect. Moreover, we give two counterexamples to show that the conclusions of their two lemmas are essentially wrong, which means they did not provide a proof for the first part of the conjecture.