The Power Graph of a Dihedral Group

Authors

  • Evi Yunartika Asmarani Universitas Mataram
  • Abdul Gazir Syarifudin Universitas Mataram
  • I Gede Adhitya Wisnu Wardhana Universitas Mataram
  • Ni Wayan Switrayni Universitas Mataram

DOI:

https://doi.org/10.29303/emj.v4i2.117

Abstract

Graph theory is one of the topics in mathematics that is quite interesting to study because it is applicable and can be combined with other mathematical topics such as group theory. The combination of graph theory and group theory is that graphs can be used to represent a group. An example of a graph is a power graph. A power graph of the group  is defined as a graph whose vertex set is all elements of  and two distinct vertices  and  are connected if and only if  or for a positive integer x and y. In this study, the author discusses the power graph of the dihedral group  The results obtained from this study are the power graph of the dihedral group  where  with  prime numbers and an  natural number is a graph consisting of two non-disjoint subgraphs, namely complete subgraphs and star subgraphs. And we find that its radius and diameter are 1 and 2.

References

Abdussakir. (2017). Radius, Diameter, Multiplisitas Sikel, dan Dimensi Metrik Graf Komuting dari Grup Dihedral. Jurnal Matematika “Mantik”, 3 (1), pp. 1-4.

Chakrabarty, I., Ghosh, S., and Sen, M.K. (2009). Undirected power graphs of semigroups. Semigroup Forum, 78(3), pp.410-426.

Chelvam, T.T. and Sattanathan, M. (2013). Power graph of finite abelian groups. Algebra and Discrete Mathematics, 16(1), pp.33-41.

Dummit, S. D., Foote, M. R. (2004). Abstract Algebra Third Edition, New York:John Wiley & Sons, Inc.

Fraleigh, J. B. (2014). A First Course in Abstract Algebra Seventh Edition. United States of America: Pearson Education Limited.

Gazir, A. S., Wardhana, I G. A. W., 2019. Subgrup Nontrivial dari Grup Dihedral. Eigen Mathematic Journal, Vol.2 No.2.

Juliana, R., Masriani, Wardhana, I.G.A.W., Switrayni, N.W., and Irwansyah, 2020, Coprime graph of integers modulo n group and its subgroups, Journal of Fundamental Mathematics and Applications (JFMA) 3 (1), 15-18

Kelarev, A. V. and Quinn, S.J. (2002). Directed graphs and combinatorial properties of semigroups. Journal of Algebra, 251(1), pp.16-26.

Masriani, Juliana, R., Syarifudin, A.G., Wardhana, I.G.A.W., Irwansyah, and Switrayni, N.W., 2020, Some Result of Non-Coprime Graph of Integers Modulo n Group for n a Prime Power, Journal of Fundamental Mathematics and Applications (JFMA) 3 (2), 107-111.

Misuki, W.U., Wardhana, I.G.A.W., Switrayni, N.W., and Irwansyah, 2021, Some Results of Non-Coprime Graph of The Dihedral Group D_2n for n a Prime Power, AIP Conference Proceedings 2329, 020005 (2021).

Munir, Rinaldi. (2010). Matematika Diskrit. Bandung:Informatika Bandung.

Nurhabibah, Syarifudin, A.G., and Wardhana, I.G.A.W., 2021, Some Results of The Coprime Graph of a Generalized Quaternion Group Q_4n, Indonesian Journal of Pure and Applied Mathematics 3 (1), pp. 29-33.

Syarifudin, A.G., Nurhabibah, Malik, D.P. and, Wardhana, I.G.A.W., Some characterizatsion of coprime graph of dihedral group D_2n, J. Phys.: Conf. Ser. 1722, 012051.

Syarifudin, A.G. Wardhana, I.G.A.W., Switrayni, N.W., and Aini, Q., 2021, The Clique Numbers and Chromatic Numbers of The Coprime Graph of a Dihedral Group, IOP Conference Series: Materials Science and Engineering 1115 (1), 012083.

Syarifudin A.G., and Wardhana, I.G.A.W., 2021, Beberapa Graf Khusus Dari Grup Quaternion, Eigen Mathematics Journal 4 (1), pp. 1-7.

Downloads

Published

2022-01-12

How to Cite

Asmarani, E. Y., Syarifudin, A. G., Wardhana, I. G. A. W., & Switrayni, N. W. (2022). The Power Graph of a Dihedral Group. EIGEN MATHEMATICS JOURNAL, 4(2), 80–85. https://doi.org/10.29303/emj.v4i2.117

Issue

Vol. 4 No. 2 Desember 2021

Section

Articles

License

All articles published in the Eigen Mathematics Journal will be available for free reading and downloading. The license applied to this journal is Creative Commons Attribution-Non-Commercial-Share Alike (CC BY-NC-SA).

Most read articles by the same author(s)

1 2 > >>