Double Roman domination number of the zero-divisor graphs of commutative rings
Ravindra Kumarravindra.pma15@iitp.ac.inAffiliation 1Department of MathematicsRam Sewak Singh Mahila College SitamarhiBabasaheb Bhimrao Ambedkar Bihar UniversityMuzaffarpur, Bihar, 842001, IndiaAffiliation 2Department of MathematicsIndian Institute of Technology PatnaPatna, Bihar, 801106, IndiaView full profile → , Ashutosh Singhashutosh_1921ma05@iitp.ac.inDepartment of MathematicsIndian Institute of Technology PatnaPatna, Bihar, 801106, IndiaView full profile → , *Om PrakashCorresponding authorom@iitp.ac.inDepartment of MathematicsIndian Institute of Technology PatnaPatna, Bihar, 801106, IndiaView full profile →
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- Received:
- 06 Sep 2023
- Published Online:
- 20 Feb 2025
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-1966
- Pages:
- 765–781
Abstract
Keywords
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References
[1] H. A. Ahangar, M. Chellali, and S. M. Sheikholeslami, “On the double Roman domination in graphs,” Discrete Applied Mathematics, vol. 232, pp. 1-7 (2017).
[2] S. Akbari, H. R. Maimani and S. Yassemi, “When a zero-divisor graph is planer or a complete r-partite graph,” Journal of Algebra, vol. 270, pp. 169-180 (2003).
[3] S. Akbari and A. Mohammadian, “On the zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 274, pp. 847-855 (2004).
[4] D. F. Anderson, A. Badawi, “The total graph of a commutative ring,” Journal of Algebra, vol. 320, pp. 2706-2719 (2008).
[5] D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston, “The zero-divisor graph of a commutative ring II,” Lecture Notes in Pure and Applied Mathematics, vol. 220, New York: Dekker (2001).
[6] D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 217, pp. 434-447 (1999).
[7] R. Balakrishnan and K. Ranganathan, “A Textbook of Graph Theory,” (2nd edition), Springer New York Heidelberg Dordrecht London (2012).
[8] I. Beck, “Coloring of commutative ring,” Journal of Algebra, vol. 116, pp. 208-226 (1988).
[9] R. A. Beeler, T. W. Haynes and S. T. Hedetniemi, “Double Roman domination,” Discrete Applied Mathematics, vol. 211, pp. 23-29 (2016).
[10] E. J. Cockayne, P. A. Dreyer Jr., S. M. Hedetniemi and S. T. Hedetniemi, “Roman domination in graphs,” Discrete Mathematics, vol. 278, pp. 11-22 (2004).
[11] N. Jafari-Rad, S. H. Jafari, and D. A. Mojdeh, “On domination in zero-divisor graph,” Canadian Mathematical Bulletin, vol. 56, no. 2, pp. 407-411 (2013).
[12] R. Kumar and O. Prakash, “Divisor graph of the complement of G(R),” Asian-European Journal of Mathematics, vol. 12, no. 4, pp. 1950057, (2019). DOI: 10.1142/S1793557119500578.
[13] R. Kumar and O. Prakash, “Pancyclic zero divisor graph over the ring Zni,” Discrete Mathematics, Algorithms and Applications, vol. 14, no. 8, pp. 22500495, (2022). doi: 10.1142/S1793830922500495.
[14] R. Kumar and O. Prakash, “Roman domination number of zero-divisor graphs over commutative rings,” Preprint arXiv:submit/6059076.
[15] E. A. Osba, S. Al-Addasi and N. A. Jaradeh, “Zero divisor graph for the ring of Gaussian integers modulo n,” Communications in Algebra, vol. 36, no. 10, pp. 3865-3877 (2008).
[16] C. S. ReVelle and K. E. Rosing, “Defendens imperium Romanum: a classical problem in military strategy,” The American Mathematical Monthly, vol. 107, no. 7, pp. 585-594 (2000).
[17] Z. Shao, S. M. Sheikholeslami, S. Nazari-Moghaddam and S. Wang, “Global double Roman domination in graphs,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 22, no. 1, pp. 31-44 (2019).
[18] I. Stewart, “Defend the Roman empire!,” Scientific American, vol. 281, no. 1, pp. 136-139 (1999).




