On the Randic index of graphs

For a given graph G = (V, E), the degree mean rate of an edge uv is an element of E is a half of the quotient between the geometric and arithmetic means of its end-vertex degrees d(u) and d(v). In this note, we derive tight bounds for the Randic index of G in terms of its maximum and minimum degree mean rates over its edges. As a consequence, we prove the known conjecture that the average distance is bounded above by the Randic index for graphs with order n large enough, when the minimum degree delta is greater than (approximately) Delta(1/3), where Delta is the maximum degree. As a by-product, this proves that almost all random (Erdos-Renyi) graphs satisfy the conjecture. (C) 2018 Elsevier B.V. All rights reserved.