# ALGORITHM FOR CONSTRUCTING THE TRIPLE UNIT GRAPH OF TYPE II OF RING Z_n USING PYTHON

### Abstract

*Let * * be a commutative ring with * * as the set of all unit elements in * *. This paper introduces a new graph associated with the ring * *, called the triple unit graph of type II, denoted by * * with the vertex set is * * − {0,1}. In TU2(R), two distinct vertices, * * and * *, are adjacent if there exists * * with * * and * * such that * *. This paper focuses on the algorithm for constructing * * using Python.** This research uses the literature study research method. The Python programming language can be used to observe the characteristic result of the graph. From the patterns generated by the algorithm, some characteristics of * * are obtained. For example, if * * is a prime and * *, then * * is a connected graph, a complete graph, a regular graph, and a Hamiltonian graph*

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### References

I. Beck, “Coloring of Commutative Rings,” J Algebra, vol. 116, no. 1, pp. 208–226, Jul. 1988, doi: 10.1016/0021-8693(88)90202-5.

D. D. Anderson and M. Naseer, “Beck’s Coloring of a Commutative Ring,” J Algebra, vol. 159, no. 2, pp. 500–514, Aug. 1993, doi: 10.1006/jabr.1993.1171.

D. F. Anderson and P. S. Livingston, “The Zero-Divisor Graph of a Commutative Ring,” J Algebra, vol. 217, no. 2, pp. 434–447, Jul. 1999, doi: 10.1006/jabr.1998.7840.

S. Akbari, D. Kiani, F. Mohammadi, and S. Moradi, “The Total Graph And Regular Graph of a Commutative Ring,” J Pure Appl Algebra, vol. 213, no. 12, pp. 2224–2228, Dec. 2009, doi: 10.1016/j.jpaa.2009.03.013.

H. Mariiaa, V. Y. Kurniawan, and Sutrima, “Zero Annihilator Graph of Semiring of Matrices Over Boolean Semiring,” in AIP Conference Proceedings, Oct. 2021, p. 020012. doi: 10.1063/5.0039557.

A. Rahmawati, V. Y. Kurniawan, and S. Wibowo, “sThe Diameter of Annihilator Ideal Graph of ℤn,” in AIP Conference Proceedings, Oct. 2021, p. 020020. doi: 10.1063/5.0039847.

E. A. Osba, S. Al-Addasi, and B. Al-Khamaiseh, “Some Properties of the Zero-Divisor Graph for the Ring of Gaussian Integers Modulo n,” Glasgow Mathematical Journal, vol. 53, no. 2, pp. 391–399, May 2011, doi: 10.1017/S0017089511000024.

R. P. Grimaldi, “Graphs from Rings,” in Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989), 1990, pp. 95–103.

N. Ashrafi, H. R. Maimani, M. R. Pournaki, and S. Yassemi, “Unit Graphs Associated With Rings,” Commun Algebra, vol. 38, no. 8, pp. 2851–2871, Aug. 2010, doi: 10.1080/00927870903095574.

B. Satyanarayana, D. Srinivasulu, and M. Bhavanari, “A Note On Unit Graph of a Ring,” IJRAR- International Journal of Research and Analytical Reviews, vol. 6, no. 1, pp. 84y–87y, Mar. 2019.

B. Satyanarayana, D. Srivanasulu, and M. Bhavanari, “Unit Graph of Type-2,” TWMS J. APP. AND ENG. MATH., vol. 12, no. 2, pp. 497–504, 2022.

E. Y. ÇELİKEL, “The Triple Zero Graph of a Commutative Ring,” Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, vol. 70, no. 2, pp. 653–663, Dec. 2021, doi: 10.31801/cfsuasmas.786804.

A. Suharto, Fundamental Bahasa Pemrograman Python, 1st ed. Purbalingga: CV. Eureka Media Aksara, 2023.

N. Kansal, B. Kaur, P. Garg, and D. Sinha, “On Another Class of Strongly Perfect Graphs,” Mathematics, vol. 10, no. 12, p. 2014, Jun. 2022, doi: 10.3390/math10122014.

Y. Tian and L. Li, “Comments on the Clique Number of Zero-Divisor Graphs of ℤn,” Journal of Mathematics, vol. 2022, pp. 1–11, Mar. 2022, doi: 10.1155/2022/6591317.

T. Hartati and V. Y. Kurniawan, “Construct The Triple Nilpotent Graph of Ring Using Python,” in AIP Conference Proceedings, Oct. 2023, p. 020004. doi: 10.1063/5.0154928.

V. Y. Kurniawan, B. Purboutomo, S. Sutrima, and N. A. Kurdhi, “Connectivity of The Triple Idempotent Graph of Ring ℤn,” in (IJCSAM) International Journal of Computing Science and Applied Mathematics, Apr. 2024, pp. 34–37.

R. Meinawati, V. Y. Kurniawan, and N. A. Kurdhi, “Algorithm for Constructing Total Graph of Commutative Ring,” in JTAM (Jurnal Teori dan Aplikasi Matematika), Apr. 2024, pp. 351–363. doi: 10.31764/jtam.v8i2.19850.

P. Wulandari, V. Y. Kurniawan, and N. A. Kurdhi, “Construct The Triple Zero Graph of Ring ℤn Using Python,” in BAREKENG: Jurnal Ilmu Matematika dan Terapan, Universitas Pattimura, Mar. 2024, pp. 0507–0516. doi: 10.30598/barekengvol18iss1pp0507-0516.

Marsudi, Teori Graf, 1st ed. Malang: UB Press, 2016.

G. Chartrand and P. Zhang, A First Course in Graph Theory. Mineola: Dover Publications, 2012.

*BAREKENG: J. Math. & App.*, vol. 18, no. 3, pp. 1639-1648, Jul. 2024.

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