On the Randić index of graphs

Abstract

For a given graph G=(V,E), the degree mean rate of an edge uv∈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 Randić 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 Randić index for graphs with order n large enough, when the minimum degree δ is greater than (approximately) Δ13, where Δ is the maximum degree. As a by-product, this proves that almost all random (Erdős–Rényi) graphs satisfy the conjecture.