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Keywords:
tree; molecular tree; Sombor-index-like graph invariant; extremal value
tree; molecular tree; Sombor-index-like graph invariant; extremal value
Summary:
I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by $\mathcal {SO}_1, \mathcal {SO}_2, \dots , \mathcal {SO}_6$. 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 $\mathcal {SO}_5$ and $\mathcal {SO}_6$ 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.
References:
[1] Gutman, I.: Geometric approach to degree-based topological indices: Sombor indices. MATCH Commun. Math. Comput. Chem. 86 (2021), 11-16. MR 4773882 | Zbl 1474.92154
[2] Gutman, I.: Sombor indices-back to geometry. Open J. Discr. Appl. Math. 5 (2022), 1-5. DOI 10.30538/psrp-odam2022.0072 | MR 4471690
[3] Gutman, I., Miljković, O.: Molecules with smallest connectivity indices. MATCH Commun. Math. Comput. Chem. 41 (2000), 57-70. MR 1787632 | Zbl 1036.92043
[4] Tang, Z., Deng, H.: Molecular trees with extremal values of the second Sombor index. Available at https://arxiv.org/abs/2208.09154 (2022), 9 pages. DOI 10.48550/arXiv.2208.09154
[5] Tang, Z., Li, Q., Deng, H.: Trees with extremal values of the Sombor-index-like graph invariants. MATCH Commun. Math. Comput. Chem. 90 (2023), 203-222. DOI 10.46793/match.90-1.203T | MR 4767035 | Zbl 1519.92356
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