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Abstract
This study investigates order divisor graphs' structural and topological properties derived from cyclic groups. Focusing on the relationship between group order and graph topology, we explore key indices, including the Wiener index, the Harary index, the first Zagreb index, and the second Zagreb index. We use a case-based approach to analyze graphs for cyclic groups of varying orders, from prime powers to more general composite structures. This work extends the theoretical framework of order divisor graphs and provides explicit formulations for their topological indices, highlighting the interplay between algebraic and graph-theoretic properties. These findings contribute to the broader understanding of algebraic graph theory and its applications.
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References
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- H. Deng, S. Balachandran, S. Elumalai, and T. Mansour, “Harary index of bipartite graphs.,” Electronic Journal of Graph Theory & Applications, vol. 7, no. 2, 2019. https://dx.doi.org/10.5614/ejgta.2019.7.2.12.
- M. Eliasi, G. Raeisi, and B. Taeri, “Wiener index of some graph operations,” Discrete Applied Mathematics, vol. 160, no. 9, pp. 1333–1344, 2012. https://doi.org/10.1016/j.dam.2012.01.014.
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- S. Zahidah, D. M. Mahanani, and K. L. Oktaviana, “Connectivity indices of coprime graph of generalized quaternion group,” J. Indones. Math. Soc, vol. 27, no. 3, pp. 285–296, 2021. https://doi.org/10.22342/jims.27.3.1043.285-296.
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References
P. Cayley, “Desiderata and suggestions: No. 2. the theory of groups: graphical representation,” American journal of mathematics, vol. 1, no. 2, pp. 174–176, 1878. https://www.jstor.org/stable/2369306.
J. Williams, A. Baig, M. Imran, and Z. Khan, “Prime graph components of finite groups,” Journal of algebra, vol. 69, pp. 587–513, 1981.
A. Abdollahi, S. Akbari, and H. Maimani, “Non-commuting graph of a group,” Journal of algebra, vol. 298, no. 2, pp. 468–492, 2006. https://doi.org/10.1016/j.jalgebra.2006.02.015.
F. Mansoori, A. Erfanian, and B. Tolue, “Non-coprime graph of a finite group,” in AIP Conference Proceedings, vol. 1750, AIP Publishing, 2016. https://doi.org/10.1063/1.4954605.
S. U. Rehman, A. Q. Baig, M. Imran, and Z. U. Khan, “Order divisor graphs of finite groups,” An. St. Ovidus Constanta, vol. 26, no. 3, pp. 29–40, 2018. https://doi.org/10.2478/auom-2018-0031.
B. ShekinahHenry and Y. I. Sheela, “Signed total domination number of order divisor graphs of some finite groups,” Recent Trends and Techniques in, p. 341, 2021.
M. Imran, S. Bibi, R. Gull, et al., “Generalized order divisor graphs associated with finite groups,” Algebra Lett., vol. 2022, pp. Article–ID, 2022. https://scik.org/index.php/abl/article/view/7630.
H. Deng, S. Balachandran, S. Elumalai, and T. Mansour, “Harary index of bipartite graphs.,” Electronic Journal of Graph Theory & Applications, vol. 7, no. 2, 2019. https://dx.doi.org/10.5614/ejgta.2019.7.2.12.
M. Eliasi, G. Raeisi, and B. Taeri, “Wiener index of some graph operations,” Discrete Applied Mathematics, vol. 160, no. 9, pp. 1333–1344, 2012. https://doi.org/10.1016/j.dam.2012.01.014.
I. Gutman and K. C. Das, “The first zagreb index 30 years after,” MATCH Commun. Math. Comput. Chem, vol. 50, no. 1, pp. 83–92, 2004.
S. Zahidah, D. M. Mahanani, and K. L. Oktaviana, “Connectivity indices of coprime graph of generalized quaternion group,” J. Indones. Math. Soc, vol. 27, no. 3, pp. 285–296, 2021. https://doi.org/10.22342/jims.27.3.1043.285-296.
J. A. Gallian, Contemporary Abstract Algebra SEVENTH EDITION. Brooks/Cole, Cengage Learning, 2010.