Equitable graphs of type Ⅱ from groups: Studying and analyzing

Alaa Altassan; Anwar Saleh; Hanaa Alashwali; Marwa Hamed; Najat Muthana; Alaa Altassan; Anwar Saleh; Hanaa Alashwali; Marwa Hamed; Najat Muthana

doi:10.3934/era.2025301

Electronic Research Archive

2025, Volume 33, Issue 11: 6805-6843. doi: 10.3934/era.2025301

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Research article

Equitable graphs of type Ⅱ from groups: Studying and analyzing

1.
Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia

Received: 06 April 2025 Revised: 28 September 2025 Accepted: 20 October 2025 Published: 17 November 2025

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)| \leq \mathrm{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.
- finite group,
- equitable graph,
- entire topological indices,
- isomorphism
Citation: Alaa Altassan, Anwar Saleh, Hanaa Alashwali, Marwa Hamed, Najat Muthana. Equitable graphs of type Ⅱ from groups: Studying and analyzing[J]. Electronic Research Archive, 2025, 33(11): 6805-6843. doi: 10.3934/era.2025301

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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)| \leq \mathrm{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.

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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 10.1142/S0219498815500656
[7]	S. U. Rehman, A. Q. Baig, M. Imran, Z. U. Khan, Order divisor graphs of finite groups, An. Stiint. Univ. Ovidius Constanta Ser. Mat., 26 (2018), 29–40. https://doi.org/10.2478/auom-2018-0031 doi: 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 doi: 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 doi: 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 doi: 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 doi: 10.1080/09728600.2021.1953359
[13]	R. Balakrishnan, K. Ranganathan, A Textbook of Graph Theory, 2$^{nd}$ edition, Springer, 2012. https://doi.org/10.1007/978-1-4614-4529-6
[14]	T. W. Hungerford, Algebra, 1$^{st}$ 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 10.2307/2313001

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