THE POWER GRAPH REPRESENTATION FOR INTEGER MODULO GROUP WITH POWER PRIME ORDER
There are many applications of graphs in various fields. Starting from chemical problems, such as the molecular shape of a compound to internet network problems, we can also use graphs to depict the abstract concept of a mathematical structure.. Groups in Algebra can be represented as a graph. This is interesting because Groups are abstract objects in mathematics. The graph of a group shows the physical form of the group by looking at the relationship between its elements. So, we can know the distance of the elements. In 2013, Abawajy et al. conducted studies related to power graphs. Power graph representation of groups of integers modulo with the order of prime numbers has been carried out in 2022 by Syechah, et al. In this article, the author provides the properties of a power graph on a group of integers modulo with the order of powers of prime numbers.
A. G. Syarifudin, Nurhabibah, D. P. Malik, and I. G. A. W. dan Wardhana, “Some characterizatsion of coprime graph of dihedral group D2n,” J Phys Conf Ser, vol. 1722, no. 1, 2021, doi: 10.1088/1742-6596/1722/1/012051.
X. Ma, H. Wei, and L. Yang, “The Coprime graph of a group,” International Journal of Group Theory, vol. 3, no. 3, pp. 13–23, 2014, doi: 10.22108/ijgt.2014.4363.
N. Nurhabibah, A. G. Syarifudin, and I. G. A. W. Wardhana, “Some Results of The Coprime Graph of a Generalized Quaternion Group Q_4n,” InPrime: Indonesian Journal of Pure and Applied Mathematics, vol. 3, no. 1, pp. 29–33, 2021, doi: 10.15408/inprime.v3i1.19670.
M. R. Gayatri, Q. Aini, Z. Y. Awanis, S. Salwa, and I. G. A. W. Wardhana, “The Clique Number and The Chromatics Number Of The Coprime Graph for The Generalized Quarternion Group,” JTAM (Jurnal Teori dan Aplikasi Matematika), vol. 7, no. 2, pp. 409–416, 2023, doi: 10.31764/jtam.v7i2.13099.
N. Nurhabibah, I. G. A. W. Wardhana, and N. W. Switrayni, “NUMERICAL INVARIANTS OF COPRIME GRAPH OF A GENERALIZED QUATERNION GROUP,” J. Indones. Math. Soc, vol. 29, no. 01, pp. 36–44, 2023.
F. Mansoori, A. Erfanian, and B. Tolue, “Non-coprime graph of a finite group,” AIP Conf Proc, vol. 1750, no. June 2016, 2016, doi: 10.1063/1.4954605.
Nurhabibah, D. P. Malik, H. Syafitri, and I. G. A. W. Wardhana, “Some results of the non-coprime graph of a generalized quaternion group for some n,” AIP Conf Proc, vol. 2641, no. December 2022, p. 020001, 2022, doi: 10.1063/5.0114975.
W. U. Misuki, G. A. W. Wardhana, and N. W. Switrayni, “Some Characteristics of Prime Cyclic Ideal On Gaussian Integer Ring Modulo,” IOP Conf Ser Mater Sci Eng, vol. 1115, no. 1, p. 012084, 2021, doi: 10.1088/1757-899X/1115/1/012084.
D. S. Ramdani, I. G. A. W. Wardhana, and Z. Y. Awanis, “THE INTERSECTION GRAPH REPRESENTATION OF A DIHEDRAL GROUP WITH PRIME ORDER AND ITS NUMERICAL INVARIANTS,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16, no. 3, pp. 1013–1020, Sep. 2022, doi: 10.30598/barekengvol16iss3pp1013-1020.
S. Akbari, F. Heydari, and M. Maghasedi, “The intersection graph of a group,” J Algebra Appl, vol. 14, no. 5, Jun. 2015, doi: 10.1142/S0219498815500656.
N. Nurhabibah, A. G. Syarifudin, I. G. A. W. Wardhana, and Q. Aini, “The Intersection Graph of a Dihedral Group,” Eigen Mathematics Journal, vol. 4, no. 2, pp. 68–73, 2021, doi: 10.29303/emj.v4i2.119.
J. Abawajy, A. Kelarev, and M. Chowdhury, “Power Graphs: A Survey,” 2013. [Online]. Available: www.ejgta.org
E. Y. Asmarani, A. G. Syarifudin, I. G. A. W. Wardhana, and N. W. Switrayni, “The Power Graph of a Dihedral Group,” Eigen Mathematics Journal, vol. 4, no. 2, pp. 80–85, 2021, doi: 10.29303/emj.v4i2.117.
B. N. Syechah, E. Y. Asmarani, A. G. Syarifudin, D. P. Anggraeni, and I. G. A. W. W. Wardhana, “Representasi Graf Pangkat Pada Grup Bilangan Bulat Modulo BerordeBilanganPrima,” Evolusi: Journal of Mathematics and Sciences, vol. 6, no. 2, pp. 99–104, 2022.
T. Chelvam and M. Sattanathan, “Power graph of finite abelian groups,” Algebra and Discrete Mathematics, vol. 16, no. 1, pp. 33–41, 2013.
J. H. Silverman, Abstract Algebra An Integrated Approach. 2022.
A. G. Syarifudin, I. G. A. W. Wardhana, N. W. Switrayni, and Q. Aini, “The Clique Numbers and Chromatic Numbers of The Coprime Graph of a Dihedral Group,” IOP Conf Ser Mater Sci Eng, vol. 1115, no. 1, p. 012083, 2021, doi: 10.1088/1757-899x/1115/1/012083.
R. G. Godsil C., Algebraic graph theory. Springer, 2001.
M. N. Husni, H. Syafitri, A. M. Siboro, A. G. Syarifudin, Q. Aini, and I. G. A. W. Wardhana, “THE HARMONIC INDEX AND THE GUTMAN INDEX OF COPRIME GRAPH OF INTEGER GROUP MODULO WITH ORDER OF PRIME POWER,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16, no. 3, pp. 961–966, Sep. 2022, doi: 10.30598/barekengvol16iss3pp961-966.
J. P. Mazorodze, S. Mukwembi, and T. Vetrík, “The Gutman index and the edge-Wiener index of graphs with given vertex-connectivity,” DiscussionesMathematicae - Graph Theory, vol. 36, no. 4, pp. 867–876, 2016, doi: 10.7151/dmgt.1900.
N. I. Alimon, N. H. Sarmin, and A. Erfanian, “The Szeged and Wiener indices for coprime graph of dihedral groups,” in AIP Conference Proceedings, American Institute of Physics Inc., Oct. 2020. doi: 10.1063/5.0018270.
Copyright (c) 2023 Lalu Riski Wirendra Putra, Zata Yumni Awanis, Salwa Salwa, Qurratul Aini, I Gede Adhitya Wisnu Wardhana
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Authors who publish with this Journal agree to the following terms:
- Author retain copyright and grant the journal right of first publication with the work simultaneously licensed under a creative commons attribution license that allow others to share the work within an acknowledgement of the work’s authorship and initial publication of this journal.
- Authors are able to enter into separate, additional contractual arrangement for the non-exclusive distribution of the journal’s published version of the work (e.g. acknowledgement of its initial publication in this journal).
- Authors are permitted and encouraged to post their work online (e.g. in institutional repositories or on their websites) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published works.