Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants \(\cal{SO}_5\) and \(\cal{SO}_6\)

I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by \(\cal{SO}_1, \cal{SO}_2, \dots, \cal{SO}_6\) $${\mathcal{S}\mathcal{O}}_1,{\mathcal{S}\mathcal{O}}_2,\dots ,{\mathcal{S}\mathcal{O}}_6$$ . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal values of the graph invariants \(\cal{SO}_5\) $${\mathcal{S}\mathcal{O}}_5$$ and \(\cal{SO}_6\) $${\mathcal{S}\mathcal{O}}_6$$ among all trees and molecular trees of order n, and characterize the trees and molecular trees that achieve the extremal values, respectively. Thus, the problem is completely solved.