Complete solution to a conjecture on the Randic index of triangle-free graphs

The Randic index R(G) of a graph G is defined by R(G) = Sigma(uv)(d(u)d(v))(-1/2) where d(u) is the degree of a vertex u in G and the summation extends over all edges ut) of G. A conjecture about the Randic index says that for any triangle-free graph G of order n with minimum degree delta >= k >= 1. one has R(G) >= root k(n-k), where the equality holds if and only if G = K-k,K-n-k. In this short note we give a confirmative proof for the conjecture. (C) 2009 Elsevier B.V. All rights reserved.