Extremal Chemical Trees for a Modified Version of Sombor Index

Abstract

‎Let $G$ be a molecular graph, where $d_u$ representes the degree of vertex $u$‎, ‎and $uv$ denotes an edge connecting vertices $u$ and $v$‎. ‎A few years ago‎, ‎a new vertex-degree-based graph invariant (topological index) was introduced by Gutman‎, ‎defined as $SO(G)=\sum_{uv\in E}\sqrt{d_u^2+d_v^2}$‎, ‎called the Sombor index‎. ‎Recently‎, ‎Kulli et al‎. ‎compared several modified versions of Sombor index (Nirmala‎, ‎Sombor‎, ‎Dharwad‎, ‎and $F$-Sombor indices)‎, ‎they found that these indices are highly correlated and their values for QSPR applications are nearly the same‎. ‎Based on this study Kulli et al‎. introduced a new vertex-degree-based topological index‎, ‎which is defined as $X(G)=\sum_{uv\in E}\sqrt{d_u^k+d_v^k}$‎, ‎where $k\geq 1$ is a real number‎. ‎In this paper‎, ‎we determine the extremal chemical trees with respect to $X$ index‎.