The Randić index and the diameter of graphs

The Randić index R(G) of a graph G is defined as the sum of 1 √dudv over all edges uv of G, where du and dv are the degrees of vertices u and v, respectively. Let D(G) be the diameter of Gwhen G is connected. Aouchiche et al. (2007) [1] conjectured that among all connected graphs G on n vertices the path Pn achieves the minimum values for both R(G)/D(G) and R(G) − D(G). We prove this conjecture completely. In fact, we prove a stronger theorem: If G is a connected graph, then R(G) − 2D(G) ≥ √ 2 − 1, with equality if and only if G is a path with at least three vertices. © 2011 Elsevier B.V. All rights reserved.