A STUDY OF THE IMPACT OF BOUNDS ON TOPOLOGICAL INDICES WITH SELECTED MODEL OF THE ALBERTSON INDEX AND THE SIGMA INDEX

Authors

  • Jasem Hamoud Department of Discrete Mathematics, Moscow Institute of Physics and Technology https://orcid.org/0009-0002-0192-3627
  • Duaa Abdullah Department of Discrete Mathematics, Moscow Institute of Physics and Technology

DOI:

https://doi.org/10.25077/jmua.15.4.481-494.2026

Keywords:

Tree, Degree Sequence, Bounds, Zagreb, Topological, Isomorphic.

Abstract

In this paper, we investigate the impact of bounds on selected topological indices by examining their extreme values. We consider the Albertson index and the Sigma index as the selected models. The Sigma index is defined as the sum of the squared differences between the degrees of adjacent vertices in a graph. Additionally, we derive bounds for the minimum and maximum Albertson indices for various tree structures. We also establish upper and lower bounds involving parameters such as the maximum degree and the minimum degree.

Downloads

Download data is not yet available.

Author Biography

Jasem Hamoud, Department of Discrete Mathematics, Moscow Institute of Physics and Technology

I am Jasem Hamoud, a researcher currently affiliated with Moscow Institute of Physics and Technology. My research interests include language acquisition, Combinatorics, Number theory, Graph Theory, I am here to collaborate with fellow researchers, share insights, and expand my knowledge through community interaction. Feel free to connect, discuss, or collaborate on projects related to these fields.

References

[1] H. Abdo, S. Brandt, and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (1), 201–206 (2014). DOI: https://doi.org/10.46298/dmtcs.1263

[2] H. Abdo, D. Dimitrov, I. Gutman, Graphs with maximal $sigma$- irregularity, Discrete Appl. Math., Elsevier B.V, 2018, DOI: 10.1016/j.dam.2018.05.013. DOI: https://doi.org/10.1016/j.dam.2018.05.013

[3] A. Ali, D. Dimitrov, T. Reti, A. M. Albalahi, A. E. Hamza, Bounds and Optimal Results for the Total Irregularity Measure, arXiv:2501.01538 [math.CO] arXiv, e-prints, 2025.

[4] M. O. Albertson, The irregularity of graph, Ars Comb. 46, 219–225 (1997).

[5] E. O. D. Andriantiana and S. Wagner, Spectral moments of trees with given degree sequence, Linear Algebra Appl. 439(12), 3980–4002 (2013). DOI: https://doi.org/10.1016/j.laa.2013.10.019

[6] A. Ashrafi, A. Ghalavand, and D. Dimitrov, On The Irregularity Of Graphs Based On The Arithmetic–geometric Mean Inequality, Math. Inequal. Appl. 26(1), 151–160 (2023).

[7] F. K. Bell, A note on the irregularity of graphs, Lin. Algebra Appl. 161, 45–54 (1992). DOI: https://doi.org/10.1016/0024-3795(92)90004-T

[8] G. Chartrand and P. Zhang, A first course in graph theory, Dover Publications, 2012.

[9] L. H. Clark, L. A. Szekely, and R. C. Entringer, An inequality for degree sequences, Discrete Math. 103, 293–300 (1992). DOI: https://doi.org/10.1016/0012-365X(92)90321-6

[10] S. Dorjsembe, L. Buyantogtokh, I. Gutman, B. Horoldagva, T. Reti, Irregularity of Graphs, J. Commun. Math. Comput. Chem. 89, 371–388 (2023). DOI: https://doi.org/10.46793/match.89-2.371D

[11] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53(4), 1184–1190 (2015). DOI: https://doi.org/10.1007/s10910-015-0480-z

[12] A. Ghalavand, A. Ashrafi, and R. , Ordering of c-cyclic graphs with respect to total irregularity, Math. Inequal. Appl. 26(1), 151–160 (2023). DOI: https://doi.org/10.7153/mia-2023-26-12

[13] I. Gutman, P. Hansen, and H. Melot, Variable neighborhood search for extremal graphs 10. Comparison of irregularity indices for chemical trees, J. Chem. Inf. Model. 45, 222–230 (2005). DOI: https://doi.org/10.1021/ci0342775

[14] I. Gutman, M. Togan, A. Yurttas, A. S. Cevik, and I. N. Cangul, Inverse Problem for Sigma Index, MATCH Commun. Math. Comput. Chem. 79(2), 491–508 (2016).

[15] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total $pi$-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535–538 (1972). DOI: https://doi.org/10.1016/0009-2614(72)85099-1

[16] I. Gutman, B. Ruscc, N. Trinajstic, and C. F. Wilcox, Graph theory and molecular orbitals. 12. Acyclic polyenes, J. Chem. Phys. 62, 3399–3405 (1975). DOI: https://doi.org/10.1063/1.430994

[17] J. Hamoud, D. Abdullah, Topological Indices with Degree Sequence D of Tree, arXiv:2503.12909 [math.CO], arXiv, e-prints, 2025.

[18] T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc., 1998.

[19] Z. Lina, T. Zhoub, and L. Miaob, The general Albertson irregularity index of graphs, AIMS Math. 7(1), 267–282 (2024). DOI: https://doi.org/10.3934/math.2022002

[20] J. B. Liu, C. Wang, and S. Wang, Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs, Bull. Malays. Math. Sci. Soc. 42, 67–78 (2019). DOI: https://doi.org/10.1007/s40840-017-0463-2

[21] M. Molloy and B. Reed, A critical point for random graphs with a given degree sequence, Random Struct. Algorithms. 6(2-3), 161–180 (1995). DOI: https://doi.org/10.1002/rsa.3240060204

[22] R. Nasiri, G. H. Fath Tabar, and I. Gutman, Extremely Irregular Trees, Bull. Acad. Sci. Serbe, Classe Sci. Math. Nat. 38, 2013.

[23] S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, The Zagreb indices 30 years after, Croat. Chem. Acta. 76(2), 113–124 (2003).

[24] J. Yang and H. Deng, The Extremal Sigma Index of Connected Graphs with Given Number of Pendant Vertices, Research Square, 2023. DOI: https://doi.org/10.21203/rs.3.rs-2646370/v1

[25] X. M. Zhang, X. D. Zhang, D. Gray, and H. Wang, The number of subtrees of trees with given degree sequence, J. Graph Theory. 73(3), 280–295 (2013). DOI: https://doi.org/10.1002/jgt.21674

Downloads

Published

22-10-2026

How to Cite

Hamoud, J., & Abdullah, D. (2026). A STUDY OF THE IMPACT OF BOUNDS ON TOPOLOGICAL INDICES WITH SELECTED MODEL OF THE ALBERTSON INDEX AND THE SIGMA INDEX. Jurnal Matematika UNAND, 15(4), 481–494. https://doi.org/10.25077/jmua.15.4.481-494.2026

Issue

Vol. 15 No. 4 (2026)

Section

Articles

License

Copyright (c) 2026 Departemen Matematika dan Sains Data FMIPA UNAND

Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

All articles published in Jurnal Matematika UNAND (JMUA) are open access and licensed under the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). This license allows published content to be freely accessed, shared, redistributed, and adapted for any purpose, including commercial use, provided that the applicable license conditions are followed.

Copyright Holder

The copyright of articles published in Jurnal Matematika UNAND is held by the Departemen Matematika dan Sains Data, Fakultas Matematika dan Ilmu Pengetahuan Alam (FMIPA), Universitas Andalas (UNAND), in accordance with the journal's stated copyright policy.

Author Rights

Under the CC BY-SA 4.0 license, authors may share, distribute, and reuse their published articles, subject to the applicable license conditions. These rights include:

  • Self-Archiving: Authors may deposit their published articles in institutional repositories, personal websites, or academic platforms.

  • Reuse: Authors may reuse their published work in subsequent publications, research, teaching materials, or other lawful purposes, provided that appropriate attribution is maintained.

  • Distribution: Authors may share and distribute their published articles without requiring additional permission under the terms of the license.

Creative Commons License (CC BY-SA 4.0)

Under the CC BY-SA 4.0 license, users are free to:

  • Share: Copy and redistribute the material in any medium or format.

  • Adapt: Remix, transform, and build upon the material for any purpose, including commercial purposes.

These permissions are subject to the following conditions:

  • Attribution: Users must give appropriate credit to the original author(s), retain relevant copyright notices, provide a link to the license, and indicate if changes were made. Attribution must not imply endorsement by the author(s), copyright holder, or journal.

  • ShareAlike: If users remix, transform, or build upon the material, they must distribute their contributions under the same CC BY-SA 4.0 license or a compatible license.

  • No Additional Restrictions: Users may not apply legal terms or technological measures that legally restrict others from exercising the rights granted by the license.

For more information about the CC BY-SA 4.0 license, please visit:

https://creativecommons.org/licenses/by-sa/4.0/

Third-Party Content

Third-party materials, including figures, tables, photographs, and illustrations, are covered by the article's CC BY-SA 4.0 license only when their inclusion under that license is authorized.

Materials excluded from the license must be clearly identified with appropriate copyright notices and attribution. Authors are responsible for obtaining any necessary permissions, while users must comply with the applicable licensing terms or permissions when reusing excluded materials.

Copyright and Licensing Information

Copyright and licensing information should be clearly displayed on each published article's landing page and included in the full-text versions (HTML and PDF), where applicable, to ensure transparency regarding copyright ownership and permitted reuse.