Geodesic
Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants $\mathcal {SO}_5$ and $\mathcal {SO}_6$
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 927-941
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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.
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.
DOI : 10.21136/CMJ.2024.0221-24
Classification : 05C09, 05C50, 05C92
Keywords: tree; molecular tree; Sombor-index-like graph invariant; extremal value 
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Gao, Wei. Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants $\mathcal {SO}_5$ and $\mathcal {SO}_6$. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 3, pp. 927-941. doi: 10.21136/CMJ.2024.0221-24

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[3] Gutman, I., Miljković, O.: Molecules with smallest connectivity indices. MATCH Commun. Math. Comput. Chem. 41 (2000), 57-70. | MR | JFM

[4] Tang, Z., Deng, H.: Molecular trees with extremal values of the second Sombor index. Available at , 9 pages. | arXiv | DOI

[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 | MR | JFM

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