Domination Parameters of Unit Graphs of Rings
Abstract
1. Introduction
1.1. Unit Graphs of Rings
1.2. The Domination Number of Unit Graphs of Rings
1.3. Layout of the Paper
2. Preliminaries
2.1. Some Notions of Graphs
- A bipartite graph is one whose vertex set can be partitioned into two subsets X and Y, so that each edge has one end in X and one end in Y, such a partition is called a bipartition of the graph.
- The degree of a vertex v in is the number of edges of incident with v, with each loop counting as two edges.
- The open neighborhood of a vertex v in a simple graph , denoted by , is the set of all vertices of which are adjacent to v.
2.2. Some Notions of Rings
- (i)
- , where is the set of unit elements of R.
- (ii)
- for any .
2.3. Some Known Results About Unit Graphs of Rings
- (i)
- if and only if R is a field.
- (ii)
- if and only if either R is a local ring which is not a field, R is isomorphic to the products of two fields such that only one of them has characteristic 2, or , where F is a field.
- (iii)
- if and only if R is not isomorphic to the product of two fields such that only one of them has characteristic 2, and , where is a local ring with maximal ideal in such a way that , for .
- (i)
- If , then the unit graph is a -regular graph.
- (ii)
- If , then for every we have and for every we have
- (i)
- If , then is a -regular graph.
- (ii)
- If , then for every we have and for every we have
3. The Relationships Between and
- (i)
- If is not adjacent to any vertex of in , then a is not adjacent to any vertex of D in .
- (ii)
- If and is not adjacent to any vertex of in , then a is not adjacent to any vertex of D in .
- (i)
- If , then .
- (ii)
- If and , then , .
- (iii)
- If and , then .
- (iv)
- If , and , then
- (v)
- If , and , then
- (vi)
- If , and , then ,
- (vii)
- If and , then , and in particular,
- (viii)
- If and , then
- (ix)
- If and , then , and
4. The Unit Graph of the Ring
4.1. Case
4.2. Case
4.3. Case
- (i)
- If and , then
- (ii)
- If and , then
- (iii)
- If and , then
- (iv)
- If , then
- (v)
- If and , then
5. The Unit Graph of the Ring Associated with Products of Four Residue Class Fields
- (i)
- If and , then , and
- (ii)
- If and , then
- (iii)
- If , then
- (i)
- n is a prime, then and by Corollary 3 (ii).
- (ii)
- , where p is a prime, then and by Corollary 3 (vi).
- (iii)
- , where are primes, then and by Corollary 3 (ix).
- (iv)
- , where are primes, then and by Proposition 4.
- (i)
- and ;
- (ii)
- ;
- (iii)
- .
- (i)
- ;
- (ii)
- .
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
- Beck, I. Coloring of commutative rings. J. Algebra 1988, 116, 208–226. [Google Scholar] [CrossRef]
- Grimaldi, R.P. Graphs from rings. Proceedings of the 20th Southeastern Conference on Combinatorics, Graph Theory, and Computing, (Boca Raton, FL,1989). Congr. Numer. 1990, 71, 95–103. [Google Scholar]
- Atiyah, M.F.; MacDonald, I.G. Introduction To Commutative Algebra, 1st ed.; CRC Press: Boca Raton, FL, USA, 1969. [Google Scholar] [CrossRef]
- Maimani, H.R.; Pournaki, M.R.; Yassemi, S. Necessary and sufficient conditions for unit graphs to be Hamiltonian. Pacific J. Math. 2011, 249, 419–429. [Google Scholar] [CrossRef]
- Heydari, F.; Nikmehr, M.J. The unit graph of a left Artinian ring. Acta Math. Hungar. 2013, 139, 134–146. [Google Scholar] [CrossRef]
- Afkhami, M.; Khosh-Ahang, F. Unit graphs of rings of polynomials and power series. Arab. J. Math. 2013, 2, 233–246. [Google Scholar] [CrossRef]
- Li, Z.; Su, H.D. The radius of unit graphs of rings. Aims Math. 2021, 6, 11508–11515. [Google Scholar] [CrossRef]
- Su, H.D.; Tang, G.; Zhou, Y. Rings whose unit graphs are planar. Publ. Math. Debr. 2015, 86, 363–376. [Google Scholar] [CrossRef]
- Su, H.D.; Noguchi, K.; Zhou, Y. Finite commutative rings with higher genus unit graphs. J. Algebra Appl. 2015, 14, 1550002. [Google Scholar] [CrossRef]
- Su, H.D.; Wei, Y. The dimaeter of unit graphs of rings. Taiwan J. Math. 2019, 23, 1–10. [Google Scholar] [CrossRef]
- Su, H.D.; Zhou, Y. On the girth of the unit graph of a ring. J. Algebra Appl. 2014, 13, 1350082. [Google Scholar] [CrossRef]
- Ashrafi, N.; Maimani, H.R.; Pournaki, M.R.; Yassemi, S. Unit Graphs associated with rings. Commun. Algebra 2010, 38, 2851–2871. [Google Scholar] [CrossRef]
- Anderson, D.F.; Asir, T.; Badawi, A.; Chelvam, T.T. Graphs from Rings; Springer: Cham, Switzerland, 2021. [Google Scholar]
- Mekis, G. Lower bounds for the domination number and the total domination number of direct product graphs. Discret. Math. 2010, 310, 3310–3317. [Google Scholar] [CrossRef]
- Kiani, S.; Maimani, H.R.; Pournaki, M.R.; Yassemi, S. Classification of rings with unit graphs having domination number less than four. Rendiconti del Seminario Matematico Della Universit di Padova 2015, 133, 173–195. [Google Scholar] [CrossRef]
- Su, H.D.; Yang, L.Y. Domination number of unit graph of Zn. Discret. Math. Algorithms Appl. 2020, 12, 2050059. [Google Scholar] [CrossRef]
- Lloyd, E.K.; Bondy, J.A.; Murty, U.S. Graph Theory with Applications; Macmillan: London, UK, 1976; Volume 290. [Google Scholar]
- Lam, T.-Y. A First Course in Noncommutative Rings; Springer: New York, NY, USA, 1991; Volume 131. [Google Scholar]
- Hungerford, T.W. Algebra; Springer Science and Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Brualdi, R.A. Introductory Combinatorics; Pearson: London, UK, 2009. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Du, T.; Gan, A. Domination Parameters of Unit Graphs of Rings. Axioms 2025, 14, 399. https://doi.org/10.3390/axioms14060399
Du T, Gan A. Domination Parameters of Unit Graphs of Rings. Axioms. 2025; 14(6):399. https://doi.org/10.3390/axioms14060399
Chicago/Turabian StyleDu, Ting, and Aiping Gan. 2025. "Domination Parameters of Unit Graphs of Rings" Axioms 14, no. 6: 399. https://doi.org/10.3390/axioms14060399
APA StyleDu, T., & Gan, A. (2025). Domination Parameters of Unit Graphs of Rings. Axioms, 14(6), 399. https://doi.org/10.3390/axioms14060399