Trees with smaller Harmonic indices

Abstract

The harmonic index $H\left(G\right)$ 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 paper, we determine (i) the trees of order $n$ and $m$ pendant

vertices with the second smallest harmonic index, (ii) the trees of

order $n$ and diameter $r$ with the smallest and the second

smallest harmonic indices, and (iii) the trees of order $n$ with

the second, the third and the fourth smallest harmonic

index,respectively.