On Harmonic Indices of Trees, Unicyclic Graphs, and Bicyclic Graphs

Abstract

The harmonic index $H(G)$ 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 $G$. We determine the $n$-vertex trees with the second and third maximum harmonic indices for $n\ge 7$, the fourth maximum harmonic index for $n\ge 10$, and fifth maximum harmonic index for $n \geq 11\), and unicyclic graphs with the second and third maximum harmonic indices for $n\ge 5$, the fourth maximum harmonic index for $n\ge 7$, and fifth maximum harmonic index for $n\ge 8$, and bicyclic graphs with the maximum harmonic index for $n\ge 6$, the second and third maximum harmonic indices for $n\ge 7$, and fourth maximum harmonic index for $n\ge 9$.