Volume 26 • Issue 1 • PP: 64-81 • 2025
On Energy of Double Dominating Bipolar Single-Valued Neutrosophic Graph
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Domination and graph energy are fundamental concepts in graph theory for addressing unpredictable phenomena, and they have attracted considerable interest from researchers. In recent developments, the concept of dominating energy has become increasingly significant in the study of graph energies. While fuzzy graphs (FG) sometimes fall short in delivering optimal results, the neutrosophic set (NS) as well as neutrosophic graphs (NG) offer a robust alternative, effectively managing the uncertainties linked with inconsistent and indeterminate information in real-world scenarios. Most existing research on domination energy in the fuzzy environment focus solely on a single membership function. In contrast, bipolar neutrosophic models, which account for both positive and negative influences, provide a more versatile and applicable approach. This paper focuses on advancements in NG theory to address scenarios where imprecision is represented by both positive and negative membership functions. It introduces a new concept called the double dominating energy graph, relying on the currently developed bipolar single-valued neutrosophic graphs (BSVNG). The study further explores the energy of double domination within the BSVNG framework. Specifically, it develops the adjacency matrix of a dominating BSVNG, analyzes the spectrum of this matrix, and elaborates on the associated theoretical aspects using illustrative examples. Additionally, the double domination energy of BSVNG is calculated to demonstrate its applicability. At the end of this study, conclusions are drawn and avenues for future research are discussed.
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References
[1] L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, no. 3, pp. 338–353, 1965.
[2] F. Smarandache, Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis, 1998.
[3] I. Deli, M. Ali, and F. Smarandache, “Bipolar neutrosophic sets and their application based on multi-criteria decision making problems,” in 2015 Int. Conf. Adv. Mechatronic Syst. (ICAMechS), pp. 249–254, IEEE, 2015.
[4] I. Gutman, “The energy of a graph,” Ber. Math.–Statist. Sekt. Forschungsz. Graz, vol. 103, pp. 1–22, 1978.
[5] I. Gutman and B. Zhou, “Laplacian energy of a graph,” Linear Algebra Appl., vol. 414, pp. 29–37, 2006.
[6] A. Kaufmann, Introduction to the Theory of Fuzzy Subsets, Acad. Press, 1975.
[7] A. Rosenfeld, “Fuzzy graphs,” Fuzzy Sets Their Appl. Cogn. Decis. Process, pp. 77–95, 1975.
[8] N. Anjali and S. Mathew, “Energy of a fuzzy graph,” Ann. Fuzzy Math. Inform, vol. 6, no. 3, pp. 455–465, 2013.
[9] B. Praba, V. M. Chandrasekaran, and G. Deepa, “Energy of an intuitionistic fuzzy graph,” Ital. J. Pure Appl. Math., vol. 32, pp. 431–444, 2014.
[10] S. S. Basha and E. Kartheek, “Laplacian energy of an intuitionistic fuzzy graph,” Indian J. Sci. Technol., vol. 8, no. 33, pp. 1–7, 2015.
[11] M. Akram, M. Sarwar, and W. A. Dudek, “Energy of bipolar fuzzy graphs,” Stud. Fuzziness Soft Comput, vol. 401, pp. 309–347, 2021.
[12] N. Patra, S. Mondal, M. Pal, and S. Mondal, “Energy of interval-valued fuzzy graphs and its application in ecological systems,” J. Appl. Math. Comput, pp. 1–19, Nov. 2021.
[13] A. Somasundaram, “Domination in products of fuzzy graphs,” Int. J. Uncertainty, Fuzziness Knowl.-Based Syst., vol. 13, no. 2, pp. 195–204, 2005.
[14] S. Ghobadi, N. D. Soner, and Q. M. Mahyoub, “Inverse dominating set in fuzzy graphs,” Proc. Jangjeon Math. Soc., vol. 11, no. 1, pp. 75–81, 2008.
[15] C. Natarajan and S. Ayyaswamy, “On strong (weak) domination in fuzzy graphs,” Int. J. Math. Comput. Sci., vol. 4, pp. 1035–1037, 2010.
[16] M. G. Karunambigai, M. Akram, K. Palanivel, and S. Sivasankar, “Domination in bipolar fuzzy graphs,” in 2013 IEEE Int. Conf. Fuzzy Syst. (FUZZ-IEEE), pp. 1–6, IEEE, 2013.
[17] D. Umamageswari, N. Vinothkumar, and P. Thangaraj, “Multiple domination in bipolar fuzzy graphs,” Int. J. Pure Appl. Math., vol. 117, no. 20, pp. 275–287, 2017.
[18] M. Akram, M. Sarwar, and W. A. Dudek, “Domination in bipolar fuzzy graphs,” Stud. Fuzziness Soft Comput, vol. 401, pp. 253–280, 2021.
[19] S. Banitalebi and R. A. Borzooei, “Neutrosophic special dominating set in neutrosophic graphs,” Neutrosophic Sets Syst., vol. 45, pp. 26–39, 2021.
[20] R. Ramya, N. Vinothkumar, and E. Karuppasamy, “Complementary domination in single-valued neutrosophic graphs,” Mater. Today: Proc., vol. 47, pp. 2076–2079, 2021.
[21] R. Hernández-Ortiz, L. P. Montejano, and J. A. Rodríguez-Velázquez, “Secure domination in rooted product graphs,” J. Combin. Optim, 2023.
[22] A. Kalimulla, R. Vijayaragavan, and S. S. Basha, “Dominating energy of operations on intuitionistic fuzzy graphs,” Int. J. Eng. Technol. (UAE), vol. 7, no. 4.10, Special Issue 10, pp. 328–335, 2018.
[23] R. Vijayaragavan, A. Kalimulla, and S. S. Basha, “Dominating Laplacian energy in products of intuitionistic fuzzy graphs,” Trends Math., pp. 603–612, 2019.
[24] M. Sarwar, M. Akram, and U. Ali, “Double dominating energy of m-polar fuzzy graphs,” J. Intell. Fuzzy Syst., vol. 38, no. 2, pp. 1997–2008, 2020..
[25] M. Akram, D. Saleem, and B. Davvaz, “Energy of double dominating bipolar fuzzy graphs,” J. Appl. Math. Comput, vol. 61, pp. 219–234, 2019.
[26] A. R. Gupta and P. S. Kumar, “Energy efficiency in fuzzy and neutrosophic graphs,” J. Comput. Theor. Nanosci, vol. 18, no. 5, pp. 1234–1240, 2021.
[27] S. N. F. Mohamad, R. Hasni, and B. Yusoff, “On dominating energy in bipolar single-valued neutrosophic graph,” Neutrosophic Sets Syst., vol. 56, pp. 119–152, 2023.
[28] Z. Shao, Z. Li, A. Peperko, J. Wan, and J. Žerovnik, "Independent rainbow domination of graphs," Bulletin of the Malaysian Mathematical Sciences Society, vol. 42, no. 2, pp. 417–435, 2019.
[29] S. Broumi, M. Talea, A. Bakali, and F. Smarandache, “On bipolar single-valued neutrosophic graphs,” J. New Theory, no. 11, pp. 84–102, 2016.
[30] K. Haghparast, J. Amjadi, M. Chellali, and S. M. Sheikholeslami, “Restrained {2}-domination in graphs,” RAIRO - Oper. Res., vol. 57, no. 5, pp. 2393–2405, 2023.
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