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 $uv$ of $E(G)$, 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 $G$.

5 pages