Bounds on Randić indices

The general Randić index of a molecular graph G is the sum of [ d ( u ) d ( v ) ] α over all edges u v ∈ G , where d ( v ) denotes the degree of the vertex v in G and α is an arbitrary number. When α = − 1 / 2 , it is called the Randić index. Delorme et al. stated a best possible lower bound on the Randić index of a triangle-free graph with given minimum degree. Their false proof was pointed out by Liu et al. In this note, we derive some sharp bounds on the general Randić index which implies their lower bound for triangle-free graphs of order n with maximum degree at most n / 4 , and also prove it for triangle-free graphs with small minimum degree.