On Global Offensive Alliance in Zero-Divisor Graphs
Abstract
:1. Introduction
2. Results
2.1. The Global Offensive Alliance Number of the Zero-Divisor Graph
- (1) for some .
- (2) , and for some prime integer p, and some positive integers n and r.
- (i)
- The inequality holds.
- (ii)
- If R has at least two units, then is an element of every alliance of .
- (i)
- We claim that the set is a global offensive alliance. Take ; if , then and (since is adjacent to ), while . On the other hand, if , then ; in this case, take such that . This shows that (since is adjacent to ), while .
- (ii)
- Let S be a global offensive alliance that does not contain the element . This implies that . In this way, is a global offensive alliance with vertices.
- (i) and , or
- (ii) .
- (1)
- ;
- (2)
- and if R is a reduced ring (i.e., a ring not containing non-zero elements x such that ).
- (i)
- ,
- (ii)
- ,
- (iii)
- (with , ),
- (iv)
- (with , ),
2.2. Rings with Small Global Offensive Alliance Number
- there exists , a pair of distinct vertices such that if then ,
- for , there are no edges connecting vertices of with vertices of ,
- if , then it is adjacent to both and .
3. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Juárez Morales, R.; Reyna Hernández, G.; Rosario Cayetano, O.; Romero Valencia, J. On Global Offensive Alliance in Zero-Divisor Graphs. Mathematics 2022, 10, 298. https://doi.org/10.3390/math10030298
Juárez Morales R, Reyna Hernández G, Rosario Cayetano O, Romero Valencia J. On Global Offensive Alliance in Zero-Divisor Graphs. Mathematics. 2022; 10(3):298. https://doi.org/10.3390/math10030298
Chicago/Turabian StyleJuárez Morales, Raúl, Gerardo Reyna Hernández, Omar Rosario Cayetano, and Jesús Romero Valencia. 2022. "On Global Offensive Alliance in Zero-Divisor Graphs" Mathematics 10, no. 3: 298. https://doi.org/10.3390/math10030298