The Harmonic Index on Bicyclic 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 establish sharp lower and upper bounds for the
harmonic index of bicyclic graphs and characterize the
corresponding extremal graphs.