SOME PROPERTIES ON COPRIME GRAPH OF GENERALIZED QUATERNION GROUPS
A coprime graph is a representation of finite groups on graphs by defining the vertex graph as an element in a group and two vertices adjacent to each other's if and only if the order of the two elements is coprime. In this research, we discuss the generalized Quaternion group and its properties. Then we discuss the properties of the coprime graph over the generalized Quaternion group by looking at its Eulerian, Hamiltonian, and Planarity sides. In general, the coprime graphs of the generalized quaternion group are not Eulerian, not Hamilton, and not planar graphs. The coprime graph of the generalized quaternion group is a planar graph if for a natural number .
W. Hamilton, “Computer codes used to generate the simulation results are available from the corresponding author.,” Philos. Mag., vol. 25, p. 489, 1844.
D. B. Sweetser, Doing Physics with Quaternions, 2005.
J. Vince, Quaternions for Computer Graphics, New York: Springer, 2011.
J. Kuipers, Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual reality, Princleton: Princleton University Press, 1999.
A. Cayley, "Desiderata and suggestions: No. 2. The Theory of groups: graphical representation," American Journal of Mathematics: In his Collected Mathematical Papers, vol. 10, pp. 403-405, 1878.
Y. Zhu, “Generalized Cayley graphs of semigroups I,” Semigroup Forum, vol. 84, p. 131–143, 2012.
Y. Luo, Y. Hao and G. T. Clarke, "On the Cayley graphs of completely simple semigroups," Semigroup Forum, vol. 82, pp. 288-295, 2011.
N. Hosseinzadeh and A. Assari, "Graph operations on Cayley graphs of semigroups," International Journal of Applied Mathematical Research, vol. 3, no. 1. , pp. 54-57, 2014.
Y. Yamasaki, “Ramanujan cayley graphs of the generalized quaternion groups and the hardy littlewood conjecture In Mathematical modelling for next-generation cryptography,,” Springer: Mathematics for Industry, vol. 29, 2018.
A. V. Kelarev and S. J. Quinn, "Directed graph and combinatorial properties of semigroups," J. Algebra, vol. 251, pp. 16-22., 2002.
I. Chakrabarty, S. Ghosh and M. K. Sen, "Undirected power graphs of semigroups," Semigroup Forum, 2009.
P. J. Camerona and S. Ghosh, "The power graph of a finite group," Discrete Mathematics, vol. 311, pp. 1220-1222, 2011.
D. Alireza, E. Ahmad and J. Abbas, "Some Results on the Power Graphs of Finite Groups," Science Asia, vol. 41, pp. 73-78, 2015.
F. Ali, S. Fatima and W. Wang, "On the power graphs of certain finite groups," Linear and Multilinear Algebra, 2020. .
P. J. Camerona, H. Guerra and S. Jurina, "The Power Graph of a Torsion-Free Groups," J Algebr Comb, vol. 49, p. 83–98, 2019.
X. Ma, H. Wei and L. Yang, "The Coprime Graph of Groups," International Journal of Group Theory, vol. 3, no. 3, pp. 13-23, 2014.
A. Sehgal, Manjeet and D. Singh, "Co-prime order graphs of finite Abelian groups and dihedral groups," Journal of Mahematics and Computer Science, vol. 23, pp. 196-202, 2021.
R. Juliana, Masriani, I. G. W. Wardhana and N. W. Switrayni, "Coprime Graph of Integer Modulo n Group and its Supgroups," Jorunal of Fundamental Mathematics and Applications, vol. 3, no. 1, pp. 15-18., 2020.
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.
K. Koh, F. Dong, K. L. Ng and E. G. Tay, Graph Theory, Singapore: World Scientific, 2015.
J. A. Bondy and U. S. R. Murty, Graduate text in mathematics: graph theory, New York: Springer, 2008.
A. Munandar, Pengantar Matematika Diskrit dan Teori Graf, Yogyakarta: Deepublish, 2022.
J. A. Gallian, Contemporary Abstract Algebra 9th Edition, Boston: Cengage Learning , 2017.
Algeboy, “www.planetmath.org,” 22 03 2013. [Online]. Available: https://www.planetmath.org/GeneralizedQuaternion Group. [Accesed 10 06 2022].
Copyright (c) 2023 Arif Munandar
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.