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

The Randić index R ( G ) of a graph G is defined by R ( G ) = ∑ u v 1 d ( u ) d ( v ) , where d ( u ) is the degree of a vertex u in G and the summation extends over all edges u v of G . Aouchiche et al. proposed a conjecture on the relationship between the Randić index and the diameter: for any connected graph on n ≥ 3 vertices with the Randić index R ( G ) and the diameter D ( G ) , R ( G ) − D ( G ) ≥ 2 − n + 1 2 and R ( G ) D ( G ) ≥ n − 3 + 2 2 2 n − 2 , with equalities if and only if G is a path. In this work, we show that this conjecture is true for trees. Furthermore, we prove that for any connected graph on n ≥ 3 vertices with the Randić index R ( G ) and the diameter D ( G ) , R ( G ) − D ( G ) ≥ 2 − n + 1 2 , with equality if and only if G is a path.