On Zero-Divisor Graphs of Zn When n Is Square-Free
Abstract
:1. Introduction
2. Key Definitions and Notations
3. Zero-Divisor Graph of Zn for Square-Free n
- 1.
- is a complete bipartite graph . In particular, if , then is a star graph and the center of =.
- 2.
- diameter, i.e., diam(.
- 3.
- radius, rad(, if and otherwise 2.
- Let us considerIf any of the primes or are 2, say, , then , and the only vertex in must have an edge with every vertex in , so is a star graph (Figure 3).
- To prove this, we consider an example as shown in Figure 2. Since is a bipartite graph, diameter and chromatic number
- Radius, rad(, is the minimum eccentricity of all the vertices of , which is 2 (Figure 2).
- and so on. After steps, we have
- .
4. Algorithm to Determine When n Is Square-Free
Algorithm 1 An algorithm to identifies the zero divisor elements of and to creates the partite structures of . |
Step 1: Find the number of nonzero zero-divisors of , which is provided by
Step 5: Partition the set of solutions into level sets . Step 6: Formation of edges, that is, connecting of vertices to obtain edges. |
5. Examples
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Equations | Solution Sets | Partition Sets | Edge Formations |
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Equations | Solution Sets | Partition Sets | Edge Formations |
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Alnefaie, K.; Gammi, N.; Rahman, S.; Ali, S. On Zero-Divisor Graphs of Zn When n Is Square-Free. Axioms 2025, 14, 180. https://doi.org/10.3390/axioms14030180
Alnefaie K, Gammi N, Rahman S, Ali S. On Zero-Divisor Graphs of Zn When n Is Square-Free. Axioms. 2025; 14(3):180. https://doi.org/10.3390/axioms14030180
Chicago/Turabian StyleAlnefaie, Kholood, Nanggom Gammi, Saifur Rahman, and Shakir Ali. 2025. "On Zero-Divisor Graphs of Zn When n Is Square-Free" Axioms 14, no. 3: 180. https://doi.org/10.3390/axioms14030180
APA StyleAlnefaie, K., Gammi, N., Rahman, S., & Ali, S. (2025). On Zero-Divisor Graphs of Zn When n Is Square-Free. Axioms, 14(3), 180. https://doi.org/10.3390/axioms14030180