Minimum bond incident degree indices of unicyclic graphs with fixed girth and maximum degree

Zhenhua Su; Zhenhua Su

doi:10.3934/math.2026814

AIMS Mathematics

2026, Volume 11, Issue 7: 20050-20067. doi: 10.3934/math.2026814

Previous Article Next Article

Research article Special Issues

Minimum bond incident degree indices of unicyclic graphs with fixed girth and maximum degree

Zhenhua Su ^{1,2
,
,}

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: 02 April 2026 Revised: 25 June 2026 Accepted: 01 July 2026 Published: 08 July 2026
MSC : 05C05, 05C09, 05C92

For a graph $ G $, its bond incident degree index is given by
$ BID = \sum\limits_{uv\in E(G)} f(d(u), d(v)), $
in which $ f(x, y) = f(y, x) $ is a real-valued symmetric function. In this paper, we establish three sufficient conditions respectively for the minimum bond incident degree ($ BID $) index of unicyclic graphs with given girth and those with given maximum degree. As applications, we verify that seven topological indices, including the recently proposed hyperbolic Sombor index and diminished Sombor index satisfy these sufficient conditions. This work provides a theoretical foundation for further investigations into the extremal values of topological indices.
- bond incident degree index,
- girth,
- maximum degree,
- unicyclic graphs
Citation: Zhenhua Su. Minimum bond incident degree indices of unicyclic graphs with fixed girth and maximum degree[J]. AIMS Mathematics, 2026, 11(7): 20050-20067. doi: 10.3934/math.2026814

Related Papers:

Abstract

For a graph $ G $, its bond incident degree index is given by

$ BID = \sum\limits_{uv\in E(G)} f(d(u), d(v)), $

in which $ f(x, y) = f(y, x) $ is a real-valued symmetric function. In this paper, we establish three sufficient conditions respectively for the minimum bond incident degree ($ BID $) index of unicyclic graphs with given girth and those with given maximum degree. As applications, we verify that seven topological indices, including the recently proposed hyperbolic Sombor index and diminished Sombor index satisfy these sufficient conditions. This work provides a theoretical foundation for further investigations into the extremal values of topological indices.

References

[1]	J. A. Bondy, U. S. R. Murty, Graph theory with applications, London: Macmillan, 1976.
[2]	R. Todeschini, V. Consonni, Handbook of molecular descriptors, Weinheim: Wiley-VCH, 2000.
[3]	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 networks, Heliyon, 11 (2025), e42280. http://dx.doi.org/10.1016/j.heliyon.2025.e42280 doi: 10.1016/j.heliyon.2025.e42280
[4]	J. Liu, M. Yu, G. Cai, J. Cao, The leader-follower coherence and spectral properties of weighted recursive networks under noise disturbance, Chaos Soliton. Fract., 208 (2026), 118060. http://dx.doi.org/10.1016/j.chaos.2026.118060 doi: 10.1016/j.chaos.2026.118060
[5]	J. Liu, K. Wang, X. Zhai, Statistical analysis and topological property of a class of fractal networks, Fractals, 34 (2026), 2640001. http://dx.doi.org/10.1142/S0218348X26400013 doi: 10.1142/S0218348X26400013
[6]	M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc., 97 (1975), 6609–6615. http://dx.doi.org/10.1021/ja00856a001 doi: 10.1021/ja00856a001
[7]	K. Aarthi, S. Elumalai, S. Balachandran, S. Mondal, On difference between atom-bond sum-connectivity index and Randić index of graphs, J. Appl. Math. Comput., 71 (2025), 2727–2748. http://dx.doi.org/10.1007/s12190-024-02339-2 doi: 10.1007/s12190-024-02339-2
[8]	F. Movahedi, Diminished Sombor index and its relationship with topological indices, Filomat, 39 (2025), 10519–10532. http://dx.doi.org/10.2298/FIL2529519M doi: 10.2298/FIL2529519M
[9]	I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 (2021), 11–16.
[10]	I. Gutman, Relating sombor and Euler indices, Vojnoteh. Glas., 72 (2024), 1–12. https://doi.org/10.5937/vojtehg72-48818 doi: 10.5937/vojtehg72-48818
[11]	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
[12]	I. Gutman, B.Furtula, M. Oz, Geometric approach to vertex-degree-based topological indices-Elliptic Sombor index, theory and applications, Int. J. Quantum Chem., 124 (2024), e27346. http://dx.doi.org/10.1002/qua.27346 doi: 10.1002/qua.27346
[13]	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
[14]	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
[15]	D. Vukičević, Q. Li, J. Sedlar, T. Došlić, Lanzhou index, MATCH Commun. Math. Comput. Chem., 80 (2018), 863–876.
[16]	A. Ali, E. Milovanović, S. Stankov, M. Matejić, I. Milovanović, Inequalities involving the harmonic-arithmetic index, Afr. Mat., 35 (2024), 46. http://dx.doi.org/10.1007/s13370-024-01183-8 doi: 10.1007/s13370-024-01183-8
[17]	E. Swartz, Tomáš Vetrík, General sum-connectivity index of unicyclic graphs with given maximum degree, Discrete Appl. Math., 366 (2025), 238-–249. http://dx.doi.org/10.1016/j.dam.2025.01.033 doi: 10.1016/j.dam.2025.01.033
[18]	T. Zhou, Z. Lin, L. Miao, The Sombor index of trees and unicyclic graphs with given maximum degree, Discre. Math. Lett., 7 (2021), 24–29. http://dx.doi.org/10.47443/dml.2021.0035 doi: 10.47443/dml.2021.0035
[19]	H. Liu, L. You, Y. Huang, Ordering chemical graphs by Sombor indices and its applications, MATCH Commun. Math. Comput. Chem., 87 (2022), 5–22. http://dx.doi.org/10.46793/match.87-1.005L doi: 10.46793/match.87-1.005L
[20]	P. Nithya, S. Elumalai, S. Balachandran, 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
[21]	Q. Cui, B. Zhao, The Lanzhou index of unicyclic graphs with fixed maximum degree, MATCH Commun. Math. Comput. Chem., 92 (2024), 689–696. http://dx.doi.org/10.46793/match.92-3.689C doi: 10.46793/match.92-3.689C
[22]	M. Chen, Y. Zhu, Extremal unicyclic graphs of Sombor index, Appl. Math. Comput., 463 (2024), 128374. http://dx.doi.org/10.1016/j.amc.2023.128374 doi: 10.1016/j.amc.2023.128374
[23]	Y. Zhang, H. Wang, S. Wang, On ABS index of unicyclic graphs with fixed diameters, Filomat, 39 (2025), 1311–1330. http://dx.doi.org/10.2298/FIL2504311Z doi: 10.2298/FIL2504311Z
[24]	K. C. Das, J. Bera, Resolving open problems on the Euler Sombor index, MATCH Commun. Math. Comput. Chem., 95 (2026), 901–918. http://dx.doi.org/10.46793/match.95-3.16625 doi: 10.46793/match.95-3.16625

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)