Extremal degree-based topological indices for trees with given segment number

Zhenhua Su; Zhenhua Su

doi:10.3934/math.20251217

AIMS Mathematics

2025, Volume 10, Issue 11: 27677-27695. doi: 10.3934/math.20251217

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Extremal degree-based topological indices for trees with given segment number

Zhenhua Su ^{1,2
,
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1.
School of Mathematics and Computational Sciences, Huaihua University, Huaihua, Hunan, China
2.
Key Laboratory of Intelligent Control Technology for Wuling-Mountain Ecological Agriculture in Hunan Province, Huaihua, Hunan, China

Received: 24 September 2025 Revised: 17 November 2025 Accepted: 20 November 2025 Published: 27 November 2025
MSC : 05C05, 05C09, 05C92

A degree-based topological index of a tree $ T $ is termed as
$ TI_f(T) = \sum\limits_{v_1v_2\in E(T)} f(d(v_1), d(v_2)), $
in which $ f(x, y) = f(y, x) $ denotes a real-valued function with $ x, y\geq 1 $. This paper mainly focuses on the extremal topological index problems for trees with a given segment number. We respectfully present the sufficient conditions for achieving the smallest and largest values of $ TI_f $, as well as depict the associated extremal graphs. As an application, it is verified that there are eight types of degree-based indices that meet these sufficient conditions, including the recently proposed Euler Sombor index and diminished Sombor index.
- extremal values,
- topological indices,
- trees,
- segment number
Citation: Zhenhua Su. Extremal degree-based topological indices for trees with given segment number[J]. AIMS Mathematics, 2025, 10(11): 27677-27695. doi: 10.3934/math.20251217

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Abstract

A degree-based topological index of a tree $ T $ is termed as

$ TI_f(T) = \sum\limits_{v_1v_2\in E(T)} f(d(v_1), d(v_2)), $

in which $ f(x, y) = f(y, x) $ denotes a real-valued function with $ x, y\geq 1 $. This paper mainly focuses on the extremal topological index problems for trees with a given segment number. We respectfully present the sufficient conditions for achieving the smallest and largest values of $ TI_f $, as well as depict the associated extremal graphs. As an application, it is verified that there are eight types of degree-based indices that meet these sufficient conditions, including the recently proposed Euler Sombor index and diminished Sombor index.

References

[1]	R. Todeschini, V. Consonni, Handbook of molecular descriptors, Wiley-VCH, Weinheim, 2000. http://dx.doi.org/10.1002/9783527613106
[2]	H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20. http://dx.doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[3]	I. Redžepović, Chemical applicability of Sombor indices, J. Serb. Chem. Soc., 86 (2021), 445–457. http://dx.doi.org/10.2298/JSC201215006R doi: 10.2298/JSC201215006R
[4]	P. Nithya, S. Elumalai, Smallest ABS index of unicyclic graphs with given girth, J. Appl. Math. Comput., 69 (2023), 3675–3692. http://dx.doi.org/10.1007/s12190-023-01898-0 doi: 10.1007/s12190-023-01898-0
[5]	M. Imran, R. Luo, M. Jamil, M. Azeem, K. M. Fahd, Geometric perspective to degree-based topological indices of supramolecular chain, Results Eng., 16 (2022), 100716. http://dx.doi.org/10.1016/j.rineng.2022.100716 doi: 10.1016/j.rineng.2022.100716
[6]	M. Ghorbani, Z. Vaziri, R. A. Ravandi, Y. Shang, The symmetric division Szeged index: A novel tool for predicting physical and chemical properties of complex networksh, Heliyon, 11 (2025), e42280. http://dx.doi.org/10.1016/j.heliyon.2025.e42280 doi: 10.1016/j.heliyon.2025.e42280
[7]	M. Imran, M. Azeem, M. K. Jamil, M. Deveci, Some operations on intuitionistic fuzzy graphs via novel versions of the Sombor index for internet routing, Granul. Comput., 9 (2024), 53. http://dx.doi.org/10.1007/s41066-024-00467-5 doi: 10.1007/s41066-024-00467-5
[8]	A. Ali, B. Furtula, I. Redžepović, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem., 60 (2022), 2081–20093. http://dx.doi.org/10.1007/s10910-022-01403-1 doi: 10.1007/s10910-022-01403-1
[9]	D. Vukičević, Q. Li, J. Sedlar, T. Došlić, Lanzhou index, MATCH Commun. Math. Comput. Chem., 80 (2018), 863–876.
[10]	T. Došlić, I. Martinjak, R. Škrekovski, S. Tipurić Spužević, I. Zubac, Mostar index, J. Math. Chem., 56 (2018), 2995–3013. https://doi.org/10.1007/s10910-018-0928-z doi: 10.1007/s10910-018-0928-z
[11]	I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 (2021), 11–16.
[12]	I. Gutman, Relating Sombor and Euler indices, Vojnoteh. Glas., 72 (2024), 1–12. http://dx.doi.org/10.5937/vojtehg72-48818 doi: 10.5937/vojtehg72-48818
[13]	Z. Tang, Y. Li, H. Deng, The Euler Sombor index of a graph, Int. J. Quantum Chem., 124 (2024), e27387. http://dx.doi.org/10.1002/qua.27387 doi: 10.1002/qua.27387
[14]	I. Gutman, B. Furtula, M. Oz, Geometric approach to vertex-degree-based topological indices-elliptic Sombor index, theory and application, Int. J. Quantum Chem., 124 (2024), e27346. http://dx.doi.org/10.1002/qua.27346 doi: 10.1002/qua.27346
[15]	J. Barman, S. Das, Geometric approach to degree-based topological index: Hyperbolic Sombor index, MATCH Commun. Math. Comput. Chem., 95 (2026), 63–94. http://dx.doi.org/10.46793/match95-1.03425 doi: 10.46793/match95-1.03425
[16]	F. Movahedi, I. Gutman, I. Redžepović, B. Furtula, Diminished Sombor index, MATCH Commun. Math. Comput. Chem., 95 (2026), 141–162. http://dx.doi.org/10.46793/match95-1.14125 doi: 10.46793/match95-1.14125
[17]	H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem., 87 (2022), 23–49. http://dx.doi.org/10.46793/match.87-1.023C doi: 10.46793/match.87-1.023C
[18]	F. Li, Q. Ye, Extremal graphs with given parameters in respect of general ABS index, Appl. Math. Comput., 482 (2024), 128974. http://dx.doi.org/10.1016/j.amc.2024.128974 doi: 10.1016/j.amc.2024.128974
[19]	S. Ahmad, K. C. Das, R. Farooq, On elliptic Sombor index with applications, Bull. Malays. Math. Sci. Soc., 48 (2025), 108. http://dx.doi.org/10.1007/s40840-025-01894-6 doi: 10.1007/s40840-025-01894-6
[20]	X. Li, J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem., 54 (2005), 195–208.
[21]	X. Li, D. Peng, Extremal problems for graphical function-indices and $f$-weighted adjacency matrix, Discrete Math. Lett., 9 (2022), 5–66. http://dx.doi.org/10.47443/dml.2021.s210 doi: 10.47443/dml.2021.s210
[22]	W. Gao, Extremal graphs with respect to vertex-degree-based topological indices for $c$-cyclic graphs, MATCH Commun. Math. Comput. Chem., 93 (2025), 549–566. http://dx.doi.org/10.46793/match.93-2.549G doi: 10.46793/match.93-2.549G
[23]	J. Du, X. Sun, On bond incident degree index of chemical trees with a fixed order and a fixed number of leaves, Appl. Math. Comput., 464 (2024), 128390. http://dx.doi.org/10.1016/j.amc.2023.128390 doi: 10.1016/j.amc.2023.128390
[24]	Z. Su. H. Deng, On the minimum vertex-degree-based topological indices of trees with given pendent vertices, MATCH Commun. Math. Comput. Chem., 95 (2026), 179-192. http://dx.doi.org/10.46793/match.95-1.17225 doi: 10.46793/match.95-1.17225
[25]	J. A. Bondy, U. S. R. Murty, Graph theory with applications, 1976.
[26]	H. Lin, On segments, vertices of degree two and the first Zagreb index of trees, MATCH Commun. Math. Comput. Chem., 72 (2014), 825–834.

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