On study of algebraic structures: multiset dimensions in zero-divisor graphs associated with rings

Nasir Ali; Hafiz Muhammad Afzal  Siddiqui; Muhammad Imran  Qureshi; Ümit  Karabıyık

doi:10.5269/bspm.76308

Nasir Ali COMSATS University Islamabad, Vehari Campus
Hafiz Muhammad Afzal Siddiqui Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan. https://orcid.org/0000-0003-1794-6460
Muhammad Imran Qureshi Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Pakistan. https://orcid.org/0000-0002-0681-6313
Ümit Karabıyık Mathematics and Computer Science, Necmettin Erbakan University, Konya, Turkey. https://orcid.org/0000-0001-7989-7321

Resumen

This article explores the multiset dimension (Mdim) in zero divisor graphs (ZD-graphs) of commutative rings R with unity. Given a finite ring , its zero divisors form the set L(R) , which defines the ZD-graph. We establish general bounds for Mdim in ZD-graphs and extend these results to various rings, including Gaussian integers, Ring of Z_n modulo n and quotient polynomial rings. Notably, we provide a complete characterization of Mdim for Z_n for all n. Our findings reveal structural patterns among rings with identical Mdim, emphasizing its role in isomorphism and enhancing the algebraic understanding of these graphs.

Descargas

La descarga de datos todavía no está disponible.

Citas

I. Beck. Coloring of commutative rings. J. Algebra 116 (1988), no. 1, 208–226.

D. Anderson and P. Livingston. The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), no. 1, 434–447.

S. P. Redmond. The zero-divisor graph of a non-commutative ring. Int. J. Commutative Rings 1 (2002), 203–211.

S. P. Redmond. An ideal-based zero-divisor graph of a commutative ring. Comm. Algebra 31 (2003), no. 9, 4425–4443.

D. D. Anderson and M. Naseer. Beck’s coloring of a commutative ring. J. Algebra 159 (1993), no. 2, 500–514.

N. Ashrafi, H. R. Maimani, M. R. Pournaki, and S. Yassemi. Unit graphs associated with rings. Comm. Algebra 38 (2010), no. 8, 2851–2871.

S. Spiroff and C. Wickham. A zero divisor graph determined by equivalence classes of zero divisors. Comm. Algebra 39 (2011), no. 7, 2338–2348.

D. F. Anderson and A. Badawi. The total graph of a commutative ring. J. Algebra 320 (2008), no. 7, 2706–2719.

Ali, N., Siddiqui, H. M. A., Riaz, M. B., Qureshi, M. I., & Akg¨ul, A. (2024). A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs. Heliyon, 10(10).

S. Pirzada. An Introduction to Graph Theory. Hyderabad, 2012.

D. B. West. Introduction to Graph Theory. Vol. 2, Prentice Hall, Upper Saddle River, 2001.

N. Ali, “Algorithm for visualization of zero divisor graphs of the ring Zn using MAPLE coding,” Open Journal of Discrete Mathematics, vol. 14, pp. 1–8, 2024.

L. Lu, L. Feng, and W. Liu. Signed zero-divisor graphs over commutative rings. Comm. Math. Stat. (2022), 1–15.

E. A. Osba, S. Al-Addasi, and N. A. Jaradeh. Zero divisor graph for the ring of Gaussian integers modulo n. Comm. Algebra 36 (2008), no. 10, 3865–3877.

R. Simanjuntak, P. Siagian, and T. Vetrik. The multiset dimension of graphs. arXiv preprint arXiv:1711.00225 (2017).

A. Kelenc, N. Tratnik, and I. G. Yero. Uniquely identifying the edges of a graph: the edge metric dimension. Discrete Appl. Math. 251 (2018), 204–220.

S. P. Redmond. The zero-divisor graph of a non-commutative ring. Int. J. Commutative Rings 1 (2002), 203–211.

S. P. Redmond. An ideal-based zero-divisor graph of a commutative ring. Comm. Algebra 31 (2003), no. 9, 4425–4443.

M. Basak, L. Saha, and K. Tiwary. Metric dimension of zero-divisor graph for the ring Zn. Int. J. Sci. Res. Math. Stat. Sci. 6 (2019), no. 6.

H. M. A. Siddiqui, A. Mujahid, M. A. Binyamin, and M. F. Nadeem. On certain bounds for edge metric dimension of zero-divisor graphs associated with rings. Math. Probl. Eng. 2021 (2021), 1–7.

S. Pirzada and M. Aijaz. Metric and upper dimension of zero divisor graphs associated to commutative rings. Acta Univ. Sapientiae, Inform. 12 (2020), no. 1, 84–101.

J. B. Liu, S. K. Sharma, V. K. Bhat, and H. Raza. Multiset and mixed metric dimension for starphene and zigzag-edge coronoid. Polycycl. Aromat. Compd. 43 (2023), no. 1, 190–204.

N. H. Bong and Y. Lin. Some properties of the multiset dimension of graphs. Electron. J. Graph Theory Appl. 9 (2021), no. 1, 215–221.

R. Alfarisi, L. Susilowati, D. Dafik, and S. Prabhu. Local multiset dimension of amalgamation graphs. F1000Research 12 (2023).

R. Alfarisi, L. Susilowati, and O. J. Dafik. On the local multiset dimension of some families of graphs. WSEAS Trans. Math. 22 (2023), 64–69.

Ali, N., Siddiqui, H. M. A., & Qureshi, M. I. (2024). On certain bounds for multiset dimensions of zero-divisor graphs associated with rings. arXiv preprint arXiv:2405.06180.

Ali, N., Siddiqui, H. M. A., & Qureshi, M. I. (2024). Exploring ring structures: multiset dimension analysis in compressed zero-divisor graphs. arXiv preprint arXiv:2405.06187.

Ali, N., Siddiqui, H. M. A., Qureshi, M. I., Abdalla, M. E. M., EL-Gawaad, N. A., & Tolasa, F. T. (2024). On Study of Multiset Dimension in Fuzzy Zero Divisor Graphs Associated with Commutative Rings. International Journal of Computational Intelligence Systems, 17(1), 298.