Zero-divisor graphs of small upper irredundance number
DOI:
https://doi.org/10.5269/bspm.47936Abstract
In this paper, we classify finite rings with upper irredundance number less than or equal to two. We note that, for such zero-divisor graphs, the upper irredundance number coincides with the independence number.
References
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2. Anderson, D. F. , Axtell, M. and Stickles, J. , Zero-divisor graphs in commutative rings, Commutative Algebra, Noetherian and Non-Noetherian Perspectives, In: Fontana M, Kabbaj SE, Olberding B, Swanson I, editors. New York, NY, USA: Springer-Verlag, pp. 23-45, 2010. https://doi.org/10.1007/978-1-4419-6990-3_2
3. Anderson, D. F. and Badawi, A., On the zero-divisor graph of a ring, Comm. Algebra 36, 3073-3092, (2008). https://doi.org/10.1080/00927870802110888
4. Anderson, D. F. and Livingston, P. S., The zero-divisor graph of a commutative ring, J. Algebra 217 , 434-447, (1999). https://doi.org/10.1006/jabr.1998.7840
5. Arumugam, S., Irredundance saturation number of a graph, Australas. J. Combin. 46, 37-49, (2010).
6. Ashrafi, N., Maimani, H. R., Pournaki, M.R. and Yassemi, S., Unit graphs associated with rings, Comm. Algebra 38 , 2851-2871, (2010). https://doi.org/10.1080/00927870903095574
7. Beck, I., Coloring of commutative rings, J. Algebra 116 , 208-226, (1988). https://doi.org/10.1016/0021-8693(88)90202-5
8. Behboodi, M. and Beyranvand, R., On the Structure of Commutative Rings with p k1 1 ...p kn n (1 ≤ ki ≤ 7) Zero-Divisors, Eur. J. Pure Appl. Math. 3(2), 303-316, (2010).
9. Chartrand, G. and Lesniak, L., Graphs and Digraphs, CRC, Fourth Edition (2005).
10. Cockayne, E. J., Hedetniemi, S. T. and Miller, D. J., Properties of hereditary hypergraphs and middle graphs, Canad. Math. Bull. 21 , 461-468, (1978). https://doi.org/10.4153/CMB-1978-079-5
11. Cockayne, E. J., Favaron, O., Payan, C. and Thomason, A. G. , Contributions to the theory of domination, independence and irredundance in graphs, Discrete Math. 33, , 249-258, (1981). https://doi.org/10.1016/0012-365X(81)90268-5
12. Fellows, M. R., Fricke, G. H., Hedetniemi, S. T. and Jacobs, D., The private neigh- bor cube, SIAM J. Discrete Math. 7 (1) , 41-47, (1994). https://doi.org/10.1137/S0895480191199026
13. Haynes, T. W., Hedetniemi, S. T. and Slater, P. J., Fundamentals of Domination in Graphs, Marcel Dekker Inc., New York (1998).
14. Jacobson, M. S. and Peters, K., Chordal graphs and upper irredundance, upper domination and independence, Discrete Math. 86, 59-69, (1990). https://doi.org/10.1016/0012-365X(90)90349-M
15. Mulay, S. B., Cycles and symmetries of zero-divisors, Comm. Algebra 30(7), 3533-3558, (2002). https://doi.org/10.1081/AGB-120004502
16. Smith, Z. O., Planar zero-divisor graphs, Int. J. Commut. Rings 2 , 177-188, (2003).
17. Wu, T. H., Yu and Lu, D., The structure of finite local principal ideal rings, Comm. Algebra 40, 4727-4738, (2012). https://doi.org/10.1080/00927872.2011.618860
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2022-01-30
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