Equitable graphs of type II from groups: Studying and analyzing

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

Algebraic graph theory explores the relationship between abstract algebra and graph theory. It uses algebraic concepts to define the structures of graphs and investigates how graph theory can characterize algebraic properties. There has been considerable scholarly interest in the connection between group-theoretic and graph-theoretic properties, especially concerning the role of symmetry in linking these two areas. This research contributes to the ongoing development of equitable graphs by introducing the concept of an equitable graph of Type II on finite groups which is defined on a group G, where two vertices x and y with different orders are adjacent if their orders are differ by at most the minimum of their orders (|o(x) − o(y)| ≤ min{o(x),o(y)}) or if one of them is the identity element. We investigate the properties of this graph for specific classes of groups, including cyclic, dihedral, and dicyclic groups. Furthermore, we derive general formulas for some degree-based indices of the equitable graph of Type II across various group families. Finally, we explore the relationship between the isomorphism of equitable graphs and the associated groups.

Keywords

finite group / equitable graph / entire topological indices / isomorphism

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] P. Cayley, Desiderata and suggestions: No. 2. The Theory of groups: Graphical representation, Am. J. Math., 1 (1878), 174-176. https://doi.org/10.2307/2369306

[2] P. J. Cameron, The power graph of a finite group, II, J. Group Theory, 13 (2010), 779-783. https://doi.org/10.1515/jgt.2010.023

[3] P. J. Cameron, S. Ghosh, The power graph of a finite group, Discrete Math., 311 (2011), 1220-1222. https://doi.org/10.1016/j.disc.2010.02.011

[4] H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra, 397 (2014), 18-31. https://doi.org/10.1016/j.jalgebra.2013.08.018

[5] R. K. Nath, J. Dutta, Spectrum of commuting graphs of some classes of finite groups, Mathematika, 33 (2017), 87-95. https://doi.org/10.11113/matematika.v33.n1.812

[6] S. Akhbari, F. Heydari, M. Maghasedi, The intersection graph of a group, J. Algebra Appl., 14 (2015), 1550065. https://doi.org/10.1142/S0219498815500656

[7] S. U. Rehman, A. Q. Baig, M. Imran, Z. U. Khan, Order divisor graphs of finite groups, An. Stint. Univ. Ovidius Constanta Ser. Mat., 26 (2018), 29-40. https://doi.org/10.2478/auom-2018-0031

[8] A. Altassan, A. Saleh, M. Hamed, N. Muthana, Characterizing finite groups through equitable graphs: A graph-theoretical approach, Mathematics, 12 (2024), 2126. https://doi.org/10.3390/math12132126

[9] F. Erdem, On the generating graphs of symmetric groups, J. Group Theory, 21 (2018), 629-649. https://doi.org/10.1515/jgth-2018-0004

[10] R. K. Nath, W. N. T. Fasfous, K. C. Das, Y. Shang, Common neighborhood energy of commuting graphs of finite groups, Symmetry, 13 (2021), 1651. https://doi.org/10.3390/sym13091651

[11] P. J. Cameron, Graphs defined on groups, Int. J. Group Theory, 11 (2022), 53-107. https://doi.org/10.22108/ijgt.2021.127679.1681

[12] A. Kumar, L. Selvaganesh, P. J. Cameron, T. T. Chelvam, Recent developments on the power graph of finite groups-A survey, AKCE Int. J. Graphs Comb., 18 (2021), 65-94. https://doi.org/10.1080/09728600.2021.1953359

[13] R. Balakrishnan, K. Ranganathan, A Textbook of Graph Theory , 2nd edition, Springer, 2012. https://doi.org/10.1007/978-1-4614-4529-6

[14] T. W. Hungerford, Algebra , 1st edition, Springer, 1974 https://doi.org/10.1007/978-1-4612-6101-8

[15] X. Ma, L. Li, On the metric dimension of the reduced power graph of a finite group, Taiwanese J. Math., 26 (2022), 1-15. https://doi.org/10.11650/tjm/210905

[16] X. Ma, Forbidden subgraphs in intersection power graphs of finite groups, Algebra Colloq., 32 (2025), 95-110. https://doi.org/10.1142/S1005386725000094

[17] X. Ma, L. Li, G. Zhong, Perfect codes in proper intersection power graphs of finite groups, Appl. Algebra Eng. Commun. Comput., 36 (2025), 543-555. https://doi.org/10.1007/s00200-023-00626-2

[18] M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math., 164 (2006), 51-229. https://doi.org/10.4007/annals.2006.164.51

[19] A. Saleh, S. Alsulami, M. Alsulami, Entire irregularity indices: A comparative analysis and applications, Mathematics, 13 (2025), 146. https://doi.org/10.3390/math13010146

[20] A. Saleh, N. Zeyada, M. S. Alharthi, Modified entire forgotten topological index of graphs: A theoretical and applied perspective, Symmetry, 17 (2025), 236. https://doi.org/10.3390/sym17020236

[21] A. Alwardi, A. Alqesmah, R. Rangarajan, I. N. Cangul, Entire Zagreb indices of graphs, Discrete Math. Algorithms Appl., 10 (2018), 1850037. https://doi.org/10.1142/S1793830918500374

[22] A. Saleh, S. H. Alsulami, On the entire harmonic index and entire harmonic polynomial of graphs, Symmetry, 16 (2024), 208. https://doi.org/10.3390/sym16020208

[23] I. N. Cangul, A. Yurttas, M. Togan, A. S. Cevik, Some formulae for the zagreb indices of graphs, in AIP Conference Proceedings, 1479 (2012), 365-367. https://doi.org/10.1063/1.4756139

[24] R. McHaffey, Isomorphism of finite abelian groups, Am. Math. Mon., 72 (1965), 48-50. https://doi.org/10.2307/2313001