The harmonic index of a graph

Abstract

The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{d(v_i)+d(v_j)}$ of all edges $v_iv_j$ of $G$, where $d(v_i)$ denotes the degree of the vertex $v_i$ in $G$. In this paper, we study how the harmonic index behaves when the graph is under perturbations. These results are used to provide a simpler method for determining the unicyclic graphs with maximum and minimum harmonic index among all unicyclic graphs, respectively. Moreover, a lower bound for harmonic index is also obtained.