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

The Randić index R ( G ) of a graph G is defined by R ( G ) = ∑ u v ( 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 u v of G . A conjecture about the Randić index says that for any triangle-free graph G of order n with minimum degree δ ≥ k ≥ 1 , one has R ( G ) ≥ k ( n − k ) , where the equality holds if and only if G = K k , n − k . In this short note we give a confirmative proof for the conjecture.