On the n-vertex trees with sixth to fifteenth maximum harmonic indices

The harmonic index of a graph G is denoted by H(G) and is defined as H(G)=∑uv∈E(G)2du+dv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(G)=\sum _{uv\in E(G)} \frac{2}{d_{u}+d_{v}}$$\end{document}, where du\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_u$$\end{document}, dv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_v$$\end{document} denote the degrees of the vertices u, v, respectively, of G and E(G) is the edge set of G. In this paper, the graphs having sixth to fifteenth maximum harmonic indices are characterized from the class of all n-vertex trees for sufficiently large n.