Extremal values of the modified Sombor index in trees

Based on Euclidean metrics, Gutman put forward a novel vertex-degree-based topological index, named the Sombor index. Later, the modified version of it—the modified Sombor index—was introduced. For a simple undirected graph $G$, the Sombor index and the modified Sombor index of $G$ are defined as $SO(G)=\sum _{uv\in E(G)}\sqrt{d_G(u{)}^2+d_G(v{)}^2}$ and ${}^{m}SO(G)=\sum _{uv\in E(G)}\frac{1}{\sqrt{d_G(u{)}^2+d_G(v{)}^2}}$, respectively, where $d_G(w)$ denotes the degree of $w$ in $G$. Extremal values of $SO$ have been intensively investigated. In this paper, we were concerned with the extremal values of the modified Sombor indices. First, we showed some graph transformations which can be used to compare the modified Sombor indices of two graphs. With these transformations, the first two maximum and minimum values of ${}^{m}SO$ among all trees of order $n$ were determined. We also characterized the unique tree that minimizes ${}^{m}SO$ among all trees with a fixed number of pendant vertices. In addition, the molecular trees with the maximum and minimum modified Sombor indices were investigated.