Zero Divisor Graph of Quotient Ring

Ayunda Faizatul Musyarrofah, Vira Hari Krisnawati, Noor Hidayat

Abstract


Recently, a lot of research has been carried out regarding graphs built from algebraic structures, including ring structures. One important example of a graph constructed from a ring is the zero divisor graph. For a commutative ring R, the zero divisor graph Γ(R) is defined as a simple graph with vertices that are non-zero zero divisors of R, and two distinct vertices are adjacent if and only if the product of the vertices is equal to zero. In this paper, we investigate the zero divisor graph of the quotient ring ℤp[x]/⟨x5⟩ with prime p. More precisely, we characterize some graph properties, including the order, size, adjacency matrix, degree, distance, diameter, girth, clique number, and chromatic number of Γ(ℤp[x]/⟨x5⟩).


Keywords


commutative ring; quotient ring; graphs properties; zero divisor graph.

Full Text:

PDF

References


N. Biggs, E. Lloyd, and R. Wilson, "Graph Theory Oxford," ed: University Press, Oxford, 1986.

M. Kahl, S. Stricker, L. Hutschenreiter, F. Bernard, and B. Savchynskyy, "Unlocking the Potential of Operations Research for Multi-Graph Matching," arXiv preprint arXiv:.18215, 2024.

I. Ali, M. Javaid, and Y. Shang, "Computing dominant metric dimensions of certain connected networks," Heliyon, vol. 10, no. 4, 2024.

S. Prakash, A. Reisizadeh, R. Pedarsani, and A. S. Avestimehr, "Coded computing for distributed graph analytics," IEEE Transactions on Information Theory, vol. 66, no. 10, pp. 6534-6554, 2020. DOI: 10.1109/TIT.2020.2999675.

S. Poulik, G. Ghorai, and Q. Xin, "Explication of crossroads order based on Randic index of graph with fuzzy information," Soft Computing, vol. 28, no. 3, pp. 1851-1864, 2024.

F. B. Farooq, S. Parveen, N. U. Hassan Awan, and R. Fanja, "Topological Descriptors and QSPR Modelling of HIV/AIDS Disease Treatment Drugs," Discrete Dynamics in Nature Society, vol. 2023, no. 1, p. 9963241, 2023.

M. Koutrouli, E. Karatzas, D. Paez-Espino, and G. A. Pavlopoulos, "A guide to conquer the biological network era using graph theory," Frontiers in Bioengineering Biotechnology, vol. 8, p. 34, 2020.

X. Qiang et al., "Implementing graph-theoretic quantum algorithms on a silicon photonic quantum walk processor," Science Advances, vol. 7, no. 9, p. eabb8375, 2021.

G. Chartrand, L. Lesniak, and P. Zhang, Graphs & digraphs. Chapman & Hall London, 1996.

I. Beck, "Coloring of commutative rings," Journal of algebra, vol. 116, no. 1, pp. 208-226, 1988.

D. F. Anderson and P. S. Livingston, "The Zero-Divisor Graph of a Commutative Ring," Journal of Algebra, vol. 217, no. 2, pp. 434-447, 1999/07/15/ 1999.

M. Axtell, J. Coykendall, and J. Stickles, "Zero-divisor graphs of polynomials and power series over commutative rings," Communications in Algebra, vol. 33, no. 6, pp. 2043-2050, 2005. DOI: https://doi.org/10.1081/AGB-200063357.

T. G. Lucas, "The diameter of a zero divisor graph," Journal of Algebra, vol. 301, no. 1, pp. 174-193, 2006. DOI: https://doi.org/10.1016/j.jalgebra.2006.01.019.

D. F. Anderson and S. B. Mulay, "On the diameter and girth of a zero-divisor graph," Journal of Pure Applied Algebra, vol. 210, no. 2, pp. 543-550, 2007.

A. Haouaoui and A. Benhissi, "Zero-divisors and zero-divisor graphs of power series rings," Ricerche di Matematica, vol. 65, pp. 1-13, 2016.

S. J. Pi, S. H. Kim, and J. W. Lim, "The zero-divisor graph of the ring of integers modulo n," Kyungpook Mathematical Journal, vol. 59, no. 4, pp. 591-601, 2019.

M. J. Park, E. S. Kim, and J. W. Lim, "The Zero-divisor Graph of ℤ n [X]," Kyungpook Mathematical Journal, vol. 60, no. 4, pp. 723-729, 2020.

C. Johnson and R. Sankar, "Graph energy and topological descriptors of zero divisor graph associated with commutative ring," Journal of Applied Mathematics Computing, vol. 69, no. 3, pp. 2641-2656, 2023.

J. Gallian, Contemporary abstract algebra. Chapman and Hall/CRC, 2021.




DOI: https://doi.org/10.18860/ca.v9i2.29008

Refbacks

  • There are currently no refbacks.


Copyright (c) 2024 Ayunda Faizatul Musyarrofah

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
Faximile (+62) 341 558933
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.