Some tight bounds for the harmonic index and the variation of the Randic index of graphs

The harmonic index H (G) and the variation of the Randic index R'(G) of a graph G are defined as the sum of the weights 2/d(u)+d(v) and 1/max{d(u), d(v)} over all the edges uv of G, respectively, where d(u) denotes the degree of'a vertex u to C. In this paper, we give some tight bounds for the harmonic index (or the variation of the Randic index, respectively) of G in terms of its maximum and minimum degree mean rates over its edges, where the degree mean rate is gamma(H)(e) = 2d(u)d(v)/(d(u)+d(v))(2) (or gamma(R')(e) = min{d(u),d(v)}/d(u)+d(v), respectively) of an (cl u +d o r edge e = uv is an element of E. We use the same method given by Dalfo (2018) recently, which proved the conjecture by Fajtlowicz that states that the average distance is bounded above by the Randic index, for graphs with order n large enough when delta is greater than (approximately) Delta(1/3), where Delta is the maximum degree. (C) 2019 Elsevier B.V. All rights reserved.