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) = ∑ uv 1 √ d(u)d(v) , where d(u) is the degree of a vertex u in G and the summation extends over all edges uv 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 2n−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. © 2011 Elsevier Ltd. All rights reserved.