Maximum values of the Sombor-index-like graph invariants of trees and connected graphs

A set of novel vertex-degree-based invariants was introduced by Gutman, denoted by \newline $S{O}_1,S{O}_2,\dots ,S{O}_6$. These invariants were constructed through geometric reasoning based on a new graph invariant framework. Motivated by proposed open problems in [Z. Tang, Q. Li, H. Deng, $\mathit{\text{Trees}}\mathit{\text{with}}\mathit{\text{Extremal}}\mathit{\text{Values}}\mathit{\text{of}}\mathit{\text{the}}\mathit{\text{Sombor-Index-Like}}\mathit{\text{Graph}}\mathit{\text{Invariants}}$, MATCH Commun. Math. Comput. Chem. $\mathbf{\text{90}}$ (2023) 203-222], we have found the maximum values of $S{O}_5$ and $S{O}_6$ in the set of molecular trees with a given number of vertices, respectively, and we have found the maximum value of $S{O}_5$ in a class of connected graphs.