Rumus Umum Pelabelan L(2, 1) pada Graf Koprima Grup Bilangan Bulat Modulo m

Siti Nur Kamila, Mohammad Nafie Jauhari

Abstract


Groups are nonempty sets equipped with binary operations and fulfill certain conditions. One example of a group is the group of integers modulo . A coprime graph of a group  is a graph whose vertices are elements in  and any two vertices  in  are directly connected if and only if the greatest common factor  of the orders x and y are relatively prime.  labeling has important applications in wireless telecommunication networks, where there are limited radio frequencies available and adjacent transmitters cannot use the same frequency. To overcome this problem, frequency assignment is done using the concept of labeling, where transmitters are symbolized as vertices connected by edges.  labeling is a graph labeling technique that assigns non-negative integer labels to graph vertices with the rule that two directly connected vertices must have a minimum label difference of two, and two vertices that are two apart must have a minimum label difference of one. The purpose of this research is to find out the general formula for the  labeling on the coprime graph of the group of integers modulo  and the general formula for the minimum value of the largest label of the  labeling. The steps taken in this research are to determine the coprime graph form of the group of integers modulo . Then label each vertex with a non-negative label. Then label each vertex with  labeling rule. From the result of this research, it can be concluded that the coprime graph of integer group modulo  has a  labeling with .


Keywords


L(2,1) Labeling; Coprime Graph; Integer Group Modulo m.

Full Text:

PDF

References


Abdusysyakir, Ketika kyai mengajar matematika. Malang: UIN-Malang Press, 2007. [2] D. Suryadi and N. Priatna, “Pengantar Dasar Teori Graf,” 2005. [3] Abdussakir, N. N. Azizah, and F. F. Nofandika, Teori Graf: Topik Dasar untuk Tugas Akhir/Skripsi. Malang: UIN-Malang Press, 2009. [4] M. Sattanathan and R. Kala, “An Introduction to Order Prime Graph,” Int. J. Contemp. Math. Sci., vol. 4, no. 10, pp. 467–474, 2009. [5] X. Ma, H. Wei, and L. Yang, “The Coprime graph of a group,” Int. J. Gr. Theory, vol. 3, no. 3, pp. 13–23, 2014. [6] H. R. Dorbidi, “A note on the coprime graph of a group,” Int. J. Gr. Theory, vol. 5, no. 4, pp. 17–22, 2016. [7] H. B. Shelash and M. Jasim, “Co-prime Graph of Finite Groups,” Order, no. April, 2021, doi: 10.13140/RG.2.2.25739.41762. [8] G. Chartrand, L. Lesniak, and P. Zhang, Graphs & digraphs. 2010. doi: 10.1201/b19731. [9] N. Irawati and R. Heri, “Pelabelan Total Titik Ajaib Pada Complete Graph,” Matematika, vol. 13, no. 3, pp. 136–142, 2010. [10] Z. Shao and R. Solis-Oba, “L (2, 1)-Labelings on the composition of n graphs,” Theor. Comput. Sci., vol. 411, no. 34–36, pp. 3287–3292, 2010, doi: 10.1016/j.tcs.2010.03.013. [11] J. A. Gallian, Contemporary Abstract Algebra. Boston: Cengage Learning, 2017. [12] G. T. Lee, Abstract Algebra An Introductory Course. Cham, Switzerland: Springer International Publishing AG, 2010. doi: 10.2307/3607096. [13] S. Fatimah, I. W. Sudarsana, and S. Musdalifah, “Pelabelan L(2,1) Pada Operasi Beberapa Kelas Graf,” J. Ilm. Mat. Dan Terap., vol. 13, no. 2, pp. 73–84, 2016, doi:10.22487/2540766x.2016.v13.i2.7207. [14] A. Lum, “Upper bounds on the L(2, 1)-labeling number of graphs with maximum degree ∆,” pp. 1–22, 2007. [15] J. R. Griggs and R. K. Yeh, “Labeling Graphs with a Condition at Distance 2,” vol. 5, no. 4, pp. 586–595, 1992. [16] Chang, G. J, and D. Kuo, “The L(2, 1)-labeling Problem On Graphs,” vol. 9, no. 2, pp. 309–316, 1996, doi: 10.1137/S0895480193245339.




DOI: https://doi.org/10.18860/jrmm.v3i6.28426

Refbacks

  • There are currently no refbacks.