The harmonic index for unicyclic graphs with given girth

Abstract

The harmonic index of a graph $G$ is defined as the sum of the 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 work, we present the minimum and maximum values of the harmonic index for unicyclic graphs with given girth, and characterize the corresponding extremal graphs.