The Harmonic Index on Unicyclic Graphs

Abstract

The Harmonic index $H(G)$ of a graph $G$ is defined as the sum of weights $\frac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we consider the Harmonic index of unicyclic graphs with a given order. We give the lower and upper bounds for Harmonic index of unicyclic graphs and characterize the corresponding extremal graphs.