A set of novel vertex-degree-based invariants, similar to the Sombor-index, was introduced by Gutman, denoted by $SO_1, SO_2, \ldots,SO_6$. 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, \textit{Trees with Extremal Values of the Sombor-Index-Like Graph Invariants}, MATCH Commun. Math. Comput. Chem. \textbf{90} (2023) 203-222], we have found the maximum values of $SO_5$ and $SO_6$ in the set of molecular trees with a given number of vertices, respectively, and we have found the maximum value of $SO_5$ in the set of connected graphs.

翻译：Gutman引入了一组基于顶点度的新不变量，类似于Sombor指标，记为$SO_1, SO_2, \ldots,SO_6$。这些不变量是通过基于新图不变量框架的几何推理构建的。受文献[Z. Tang, Q. Li, H. Deng, \textit{Trees with Extremal Values of the Sombor-Index-Like Graph Invariants}, MATCH Commun. Math. Comput. Chem. \textbf{90} (2023) 203-222]中提出的开放问题启发，我们分别找到了给定顶点数的分子树集合中$SO_5$和$SO_6$的最大值，并找到了连通图集合中$SO_5$的最大值。