On two conjectures concerning trees with maximal inverse sum indeg index

The inverse sum indeg (ISI) index of a graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V, E)$$\end{document} is defined as ISI(G)=∑vivj∈Edidj/(di+dj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ISI(G) = \sum _{v_i v_j \in E} d_i d_j/(d_i + d_j)$$\end{document}, where V={v0,v1,…,vn-1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=\{v_0, v_1, \ldots , v_{n-1}\}$$\end{document} and E are, respectively, the vertex set and edge set of G, and di\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i$$\end{document} is the degree of vertex vi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_i$$\end{document}. This topological index was shown to be well correlated with the total surface area of octane isomers. However, the problem of characterizing trees with maximal ISI index (optimal trees, for convenience) appears to be difficult. Let T be an n-vertex optimal tree. Recently, Chen et al. (Appl Math Comput 392:125731, 2021) proved some structural features of T, and proposed some problems and conjectures for further research. In particular, they conjectured that ISI(T)<2n-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ISI(T) < 2n-2$$\end{document}, and T has no vertices of degree 2 if n≥20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 20$$\end{document}. In this paper, we confirm these two conjectures.