Sharp bounds for the Randic index of graphs with given minimum and maximum degree

The Randic index of a graph G, written R(G), is the sum of 1/root d(u)d(v) over all edges uv in E(G). Let d and D be positive integers d < D. In this paper,(WT) prove that if G is a graph with minimum degree d and maximum degree D, then R(G) >= root dD/d+Dn; equality holds only when G is an n-vertex (d, D)-biregular. Furthermore, we show that if G is an n-vertex connected graph with minimum degree d and maximum degree D, then R(G) <= n/2 - Sigma(D-1)(i=d) 1/2 (1/root i - 1/root i+1)(2) : it is sharp for infinitely many n, and we characterize when equality holds in the bound. (C) 2018 Elsevier B.V. All rights reserved.