Abstract
Graph theory that can be used to describe the relationships among several individuals has numerous applications in diverse fields such as modern sciences and technology, database theory, data mining, neural networks, expert systems, cluster analysis, control theory, and image capturing. As a generalization of fuzzy set (FS) and intuitionistic fuzzy set (IFS), the concept of neutrosophic set is a more functional tool for handling indeterminate, inconsistent and uncertain information that exist in real life compared to FSs and IFSs. In this paper, we apply the graph theory to the single-valued neutrosophic sets and investigate a new kind of graph structure which is called single-valued neutrosophic graphs and is generalized the results concerning crisp graphs, fuzzy graphs and intuitionistic fuzzy graphs. Then we describe some of their theoretical properties, such as the Cartesian product, composition, union and join. By applying two different procedures to solve single-valued neutrosophic decision-making problems, a neutrosophic graph-based multicriteria decision-making model is developed to consider relationships among the multi-input arguments which cannot be handled well by means of the existing methods. Finally, two illustrative examples are given to demonstrate the applicability, feasibility, effectiveness and advantages of these two proposed approaches.
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References
Akram M Alshehri NO (2014) Intuitionistic fuzzy cycles and intuitionistic fuzzy trees. Scientific World Journal Volume 2014, Article ID 305836
Akram M, Davvaz B (2012) Strong intuitionistic fuzzy graphs. Filomat 26(1):177–195
Akram M, Dudek WA (2012) Regular bipolar fuzzy graphs. Neural Comput Appl 21(1):S197–S205
Akram M, Dudek WA (2013) Intuitionistic fuzzy hypergraphs with applications. Inform Sci 218:182–193
Akram M, Karunambigai MG, Kalaivani OK (2012) Some metric aspects of intuitionistic fuzzy graphs. World Appl Sci J 17:1789–1801
Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96
Berge C (1976) Graphs and hypergraphs. North-Holland, New York
Bhattacharya P (1987) Some remarks on fuzzy graphs. Pattern Recognit Lett 6:297–302
Bhutani KR, Battou A (2003) On M-strong fuzzy graphs. Inf Sci 155:103–109
Bhutani KR, Rosenfeld A (2003) Strong arcs in fuzzy graphs. Inf Sci 152:319–322
Biswas P, Pramanik S, Giri BC (2016) TOPSIS method for multi-attribute group decision-making under single-valued neutrosophic environment. Neural Comput Appl 27(3):727–737
Broumi S, Smarandache F (2013) Correlation coefficient of interval neutrosophic set. Appl Mech Mater 436:511–517
Broumi S, Smarandache F (2013) Several Similarity Measures of Neutrosophic Sets. Neutros Sets Syst 1:54–62
Broumi S, Smarandache F (2014) Neutrosophic refined similarity measure based on cosine function. Neutros Sets Syst 6:41–47
Chi PP, Liu PD (2013) An Extended TOPSIS Method for multiple attribute decision making problems based on interval neutrosophic set. Neutros Sets Syst 1:63–70
Diestel R (2006) Graph theory. Springer, Berlin
Kauffman A (1973) Introduction a la Theorie des Sous-emsembles Flous, Masson et Cie 1
Liu PD, Wang YM (2014) Multiple attribute decision making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Comput Appl 25(7–8):2001–2010
Liu PD, Chu YC, Li YW, Chen YB (2014) Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. Int J Fuzzy Syst 16(2):242–255
Majumdar P, Samanta SK (2014) On similarity and entropy of neutrosophic sets. J Intell Fuzzy Syst 26(3):1245–1252
Mordeson JN, Nair PS (1998) Fuzzy graphs and fuzzy hypergraphs, 2nd edn. Physica Verlag, Heidelberg
Mordeson JN, Peng CS (1994) Operations on fuzzy graphs. Inf Sci 79:159–170
Parvathi R, Thilagavathi S, Karunambigai MG (2009) Intuitionistic fuzzy hypergraphs. Cybernet Inform Technol 9:46–48
Parvathi R, Karunambigai MG, Atanassov KT (2009) Operations on intuitionistic fuzzy graphs, fuzzy systems. In: FUZZ–IEEE 2009. IEEE international conference, pp 1396–1401
Peng JJ, Wang JQ, Zhang HY, Chen XH (2014) An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl Soft Comput 25:336–346
Peng JJ, Wang JQ, Wu XH, Wang J, Chen XH (2015) Multi-valued neutrosophic sets and power aggregation operators with their applications in multi-criteria group decision-making problems. Int J Comput Intell Syst 8(2):345–363
Peng JJ, Wang JQ, Wang J, Zhang HY, Chen XH (2016) Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int J Syst Sci 47(10):2342–2358
Pramanik S, Biswas P, Giri BC (2017) Hybrid vector similarity measures and their applications to multi-attribute decision making under neutrosophic environment 28(5):1163–1176
Rosenfeld A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their applications. Academic Press, New York, pp 77–95
Şahin R (2017) Cross-entropy measure on interval neutrosophic sets and its applications in multicriteria decision making. Neural Comput Appl 28(5):1177–1187
Şahin R, Küçük A (2015) Subsethood measure for single-valued neutrosophic sets. J Intell Fuzzy Syst 29(2):525–530
Şahin R, Liu PD (2016) Maximizing deviation method for neutrosophic multiple attribute decision making with incomplete weight information. Neural Comput Appl 27(7):2017–2029
Şahin R, Liu P (2017) Correlation coefficient of single-valued neutrosophic hesitant fuzzy sets and its applications in decision making. Neural Comput Appl 28(6):1387–1395
Shannon A, Atanassov KT (1994) A first step to a theory of the intuitionistic fuzzy graphs. In: Lakov D (ed) Proceedings of FUBEST, Sofia, pp 59–61
Smarandache F (1999) A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic. American Research Press, Rehoboth
Smarandache F (2015) Symbolic neutrosophic logic. Europa Nova, Bruxelles, p 194
Sunitha MS, Vijayakumar A (2002) Complement of a fuzzy graph. Indian J Pure Appl Math 33:1451–1464
Wang H, Smarandache F, Zhang YQ, Sunderraman R (2005) Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Phoenix
Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single-valued neutrosophic sets. Multispace Multistructure 4:410–413
Ye J (2013) Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int J Gen Syst 42(4):386–394
Ye J (2014) Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making. Int J Fuzzy Syst 16(2):204–211
Ye J (2014) Single-valued neutrosophic cross-entropy for multicriteria decision making problems. Appl Math Model 38:1170–1175
Ye J (2014) A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J Intell Fuzzy Syst 26:2459–2466
Ye J (2014) Multiple attribute group decision-making method with completely unknown weights based on similarity measures under single-valued neutrosophic environment. J Intell Fuzzy Syst 27(12):2927–2935
Ye J (2015) Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artif Intell Med 63(3):171–179
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–356
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Şahin, R. An approach to neutrosophic graph theory with applications. Soft Comput 23, 569–581 (2019). https://doi.org/10.1007/s00500-017-2875-1
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DOI: https://doi.org/10.1007/s00500-017-2875-1