Note on the Harmonic Index of a Graph

Abstract

The harmonic index of a graph \(G\) is defined as the sum of weights \(\frac{2}{\deg(v) + \deg(u)}\) of all edges \(uv\) in \(E(G)\), where \(\deg(v)\) denotes the degree of a vertex \(v\) in \(V(G)\). In this note, we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561-566] and establish some upper and lower bounds on the harmonic index of \(G\).