International Journal of Pure Applied Mathematics Volume 119 No. 14 2018, 891-898 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu ON M-POLAR INTUITIONISTIC FUZZY GRAPHS K. Sankar 1, D. Ezhilmaran 2 1 Department of Mathematics, Bharathiar University, Coimbatore, Tamilnadu, India. 2 School of Advanced Sciences, VIT University, Vellore, Tamilnadu, India Abstract : In the real world scenario n (n>2) array information are required to describe a problem completely. To analyze these type of problems we use m-polar fuzzy graphs. Otherwise it cannot be represented completely using fuzzy graphs or bipolar fuzzy graphs. Though we use m-polar fuzzy graphs people may not satisfy with the available data, they always look for more better data than the available that they may have little unsatisfaction(hesitation) with the data, considering this we use m-polar intuitionistic fuzzy graphs. In this paper, some operations are defined to formulate these graphs. MSC: 05C72, 05C76 Keywords: m-polar intuitionistic fuzzy sets, Generalized m-polar intuitionistic fuzzy graphs, 1.Introduction The concept of graph is started when Euler attempted to solve the Konigsberg bridge problem in 1735. Mobius defined the complete bipartite graphs in 1840, then Kuratowski has proved they are planar graphs using recreational problems. In the present world graph theory is very much used in the field of computer science including data mining, image processing, clustering networking. Zadeh [17] introduced the fuzzy sets in 1965. It gives a considerable change in the field of science technology it initiated a new path Fuzzy Logic. Later in 1994, Zhang [20,21] initiated the concept of bipolar fuzzy sets. Juanjuan chen et al. [1] introduced m-polar fuzzy sets as generalization of bipolar fuzzy sets. Kafmann [6] introduced the definition of fuzzy graph firstly from Zadeh s fuzzy relations [17-19]. Rosenfeld [9] developed another group of elaborated definitions, including the fuzzy vertex, fuzzy edges paths, cycles, connectedness etc. Mordeson Nair [8] have studied the complement of a fuzzy graph. Fuzzy intersection graphs were studied by McAllister [7]. Samanta Pal studied fuzzy tolerance graphs [10], fuzzy threshold graphs [11], bipolar fuzzy hypergraphs [12], irregular bipolar fuzzy graphs [13], fuzzy k- competition graphs, m step fuzzy competition graphs [14,15] fuzzy planar graphs [16]. D.Ezhilmaran K.Sankar [22]have studied bipolar intuitionistic fuzzy graphs some of its properties in 2015. In 2014, Juanjuan Chen et al [1] defined m-polar fuzzy graphs. Ghorai Pal introduced some operations the density of m-polar fuzzy graphs [2], studied m-polar fuzzy planar graphs [3] defined faces the dual nature of m-polar fuzzy planar graphs [4].. In this paper the Cartesian product, composition, union join of two m-polar intuitionistic fuzzy graphs are defined. Some important properties of isomorphisms, strong m-polar intuitionistic fuzzy graphs, selfcomplementary m-polar intuitionistic fuzzy graphs self-complementary strong m-polar intuitionistic fuzzy graphs are discussed. 2. Preliminaries In this section, we briefly recall some definitions of undirected graphs, the notions of fuzzy sets, bipolar fuzzy sets, intuitionistic fuzz sets, m-polar fuzzy sets, fuzzy graph, bipolar fuzzy graph, bipolar intuitionistic fuzzy graph m-polar fuzzy graph. Definition 2.1: [5]. A graph is an ordered pair G*= (V,E), where V is the set of vertices of G* E is the set of edges of G*. Two vertices x y in an undirected graph G* are said to be adjacent in G* if {x,y} is an edge of G*. A simple graph is an undirected graph that has no loops no more than one edge between any two different vertices. Definition 2.2: [5] Consider the cartesian product, of graphs. Then,,,,,,. Definition 2.3: [5] Let,, be two simple graphs. Then the composition of graphs is denoted by =,, where,,, E is defined in. Note that. Definition 2.4: [5] The union of two simple graphs,, is the simple graph with the vertex set edge set. The union of is denoted by,. 891
International Journal of Pure Applied Mathematics Definition 2.5: [5] The complementary graph of a simple graph has the same vertices as. Two vertices are adjacent in if only if they are not adjacent in. Definition 2.6: [17] A fuzzy subset on a set X is a map 0,1. A map 0,1 is called a fuzzy relation on X if, for all x,y X. A fuzzy relation is symmetric if,, for all x,y X. Definition 2.7: [25] Let X be a non empty set. A bipolar fuzzy set B in X is an object having the form,, where 0,1 1,0 are mappings. Definition 2.8: [24] Let X be a non empty set. An intuitionistic fuzzy set B =,, Where 0,1 0,1 are mapping such that 0 1. Definition 2.9: [22] Let X be a non empty set. A bipolar intuitionistic fuzzy set B=,,,, where 0,1, 1,0 0,1, 1,0 are the mappings such that 0 1,1 0. Definition 2.10: [22] A bipolar intuitionistic fuzzy graph of a graph G * = (V,E) is a pair G(A,B) where A=μ,μ,, is a bipolar intuitionistic fuzzy set in V B = μ,μ,, is a bipolar intuitionistic fuzzy set in V V such that μ μ μ, μ μ μ,, μ μ 0, 0. Definition 2.11: [23] An m-polar fuzzy set on a set X is the mapping : 0,1 (0,1 is the m-th power of 0,1). The set of all m-polar fuzzy sets on X is denoted by m(x). Definition 2.12: [23] A generalized m-polar fuzzy graph of a graph, is a pair,, where : 0,1 is m-polar fuzzy set in V : 0,1 is an m-polar fuzzy set in such that, for all 0 for all.0 0,0,0,0 is the smallest element in 0,1. A is called the m-polar fuzzy vertex set of G B is called the m-polar fuzzy edge set of G. 3. m-polar intuitionistic fuzzy graph Definition 3.1: An m-polar intuitionistic fuzzy set on a set X is a pair of mapping, :X 0,1, gives the membership values gives non-membership values for all. Definition 3.2: An m-polar intuitionistic fuzzy graph of a graph, is an ordered triple,,where,,., are m-polar membership m-polar non-membership functions from 0,1, are m-polar membership non-membership functions from 0,1 such that, means minimum of, ] in the sense that,, means maximum of, ] in the sense that, where : 0,1 0,1is the i th projection mapping. Fig:1 Example of 3-polar intuitionistic fuzzy graph G In the graph, the vertices edges have the following membership non-membership values., 0.4,0.2,0.1, 0.3,0.4,0.3,, 0.5,0.6,0.4, 0.4,0.3,0.2,, 0.3,0.2,0.1, 0.5,0.5,0.4 a, b, 0.2,0.2,0.1, 0.6,0.5,0.4, b, c, 0.2,0.2,0.1, 0.7,0.6,0.5, 892
International Journal of Pure Applied Mathematics c, a, 0.2,0.1,0.1, 0.6,0.7,0.5 It is a 3-polar intuitionistic fuzzy graph. Definition 3.3: The Cartesion product of two m-polar intuitionistic fuzzy graphs,,,, of the graphs,,, respectively, is defined as a triple,, such that for 1,2,3. (i), = for all,. (ii),, = for all, for all. (iii),,= (iv) for all, for all. (v),, = 0 for all 0,,. Proposition 3.4: The Cartesion product =,, of two m-polar intuitionistic fuzzy graphs,,,, of the graphs,, is an m- polar intuitionistic fuzzy graph of Proof: Let,. Then for 1,2,3,, =,,.,,,,. Let,. Then for 1,2,3,,=,,. Let,. Then for 1,2,3,,,,. Let,,. Then, for 1,2.,, 0,,,, 0,, Fig:2 Example of cartesion product of 3-polar intuitionistic fuzzy graphs In the graph, vertices the edge have the following membership non-membership values, 0.4,0.2,0.1, 0.3,0.4,0.3,, 0.5,0.6,0.4, 0.4,0.3,0.2,, 0.3,0.2,0.1, 0.5,0.5,0.4 In the graph, vertices the edge have the following membership non-membership values, 0.6,0.3,0.1, 0.2,0.4,0.6,, 0.4,0.6,0.8, 0.3,0.2,0.1,, 0.3,0.2,0.1, 0.4,0.5,0.7 In the graph, vertices edges have the following membership non-membership values a, c, 0.4,0.2,0.1, 0.3,0.4,0.6, a, d, 0.4,0.2,0.1, 0.3,0.4,0.3, b, c, 0.5,0.3,0.1, 0.4,0.4,0.6, b, d, 0.4,0.6,0.4, 0.4,0.3,0.2 a, c, a, d, 0.3,0.2,0.1, 0.4,0.5,0.7, a, d, b, d, 0.3,0.2,0.1, 0.5,0.5,0.4, b, d, b, c, 0.3,0.2,0.1, 0.4,0.5,0.7, a, c, b, c, 0.3,0.2,0.1, 0.5,0.5,0.6. From this it is clear that the cartesion product of two 3- polar intuitionistic fuzzy graphs is again a 3-polar intuitionistic fuzzy graph. Definition 3.5: The composition =,, of two m-polar intuitionistic fuzzy graphs,,,, of the graphs 893
International Journal of Pure Applied Mathematics,, respectively is defined as follows: for 1,2. (i), = for all,. (ii),, = for all, for all. (iii),,= for all, for all. (iv),, = for all,,. (v),, = 0 for all 0,,. Proposition 3.6: The composition of two m- polar intuitionistic fuzzy graphs,,,, of the graphs,, respectively is an m-polar intuitionistic fuzzy graph. Proof: Let,. Then for 1,2,3,, =,,,, =,, Let,. Then for 1,2,3, the proof is similar to the above. Let,,. Therefore,. Then we have for each 1,2,3,,, =,, similarly,, =,, Hence is also an m-polar intuitionistic fuzzy graph. Fig:3 Example of composition of 3-polar intuitionistic fuzzy graphs In the graph, the vertices edges have the following membership non-membership values, 0.1,0.4,0.5, 0.3,0.2,0.1,, 0.2,0.5,0.6, 0.3,0.2,0.3,, 0.1,0.3,0.4, 0.4,0.4,0.5 In the graph, the vertices edges have the following membership non-membership values, 0.4,0.2,0.3, 0.5,0.3,0.3,, 0.3,0.4,0.5, 0.5,0.3,0.4,, 0.2,0.1,0.3, 0.6,0.4,0.5 In the graph, the vertices edges have the following membership non-membership values a, c, 0.2,0.1,0.3, 0.5,0.3,0.3, a, d, 0.1,0.4,0.5, 0.5,0.3,0.4, b, c, 0.2,0.2,0.3, 0.5,0.3,0.3, b, d, 0.2,0.4,0.5, 0.5,0.3,0.4 a, c, a, d, 0.1,0.1,0.3, 0.6,0.4,0.5, a, d, b, d, 0.1,0.3,0.4, 0.5,0.4,0.5, b, d, b, c, 0.2,0.1,0.3, 0.6,0.4,0.5, a, c, b, c, 0.1,0.2,0.3, 0.5,0.4,0.5, a, c, b, d, 0.1,0.2,0.3, 0.5,0.4,0.5, b, c, a, d, 0.1,0.2,0.3, 0.5,0.4,0.5. From this, it is clear that the composition of two 3-polar intuitionistic fuzzy graphs is also a 3-polar intuitionistic fuzzy graph. Definition 3.7: The union =,, of two m-polar intuitionistic fuzzy graphs,, respectively is defined 894
International Journal of Pure Applied Mathematics as following: for 1,2,3, = x x x x x x = xy xy xy xy xy xy 0 = if 0. Proposition 3.8: The union of two m-polar intuitionistic fuzzy graphs,,,, of the graphs,, respectively is also an m-polar intuitionistic fuzzy graph. Proof: Let. Then for 1,2,3 = = Similarly, if, then for membership for nonmembership. if, then for membership. This completes the proof. Fig:4 Example of union of 3-polar intuitionistic fuzzy graphs In the graph, the vertices edges have the following membership non-membership values., 0.3,0.4,0.2, 0.4,0.2,0.1,, 0.6,0.5,0.4, 0.2,0.3,0.1,, 0.2,0.6,0.3, 0.3,0.2,0.5, 0.3,0.7,0.8, 0.6,0.2,0.1 a, b, 0.2,0.3,0.2, 0.5,0.4,0.2, b, c, 0.2,0.4,0.2, 0.4,0.5,0.6, a, d, 0.2,0.3,0.2, 0.7,0.4,0.3, b, d, 0.2,0.3,0.4, 0.7,0.4,0.3 In the graph, the vertices edges have the following membership non-membership values., 0.3,0.2,0.5, 0.4,0.3,0.4,, 0.5,0.6,0.2, 0.3,0.2,0.1,, 0.6,0.3,0.7, 0.2,0.4,0.2, 0.3,0.4,0.5, 0.2,0.3,0.4 a, b, 0.2,0.1,0.2, 0.5,0.4,0.5, b, c, 0.4,0.2,0.1, 0.4,0.6,0.3, c, f, 0.2,0.3,0.3, 0.3,0.5,0.6, b, f, 0.2,0.3,0.1, 0.4,0.4,0.5 In the graph, the vertices edges have the following membership non-membership values., 0.3,0.4,0.5, 0.4,0.2,0.1,, 0.6,0.6,0.4, 0.2,0.2,0.1,, 0.6,0.6,0.7, 0.2,0.2,0.2, 0.3,0.7,0.8, 0.6,0.2,0.1,, 0.4,0.7,0.5, 0.2,0.3,0.4 a, b, 0.2,0.3,0.2, 0.5,0.4,0.2, b, c, 0.4,0.4,0.2, 0.4,0.5,0.3, a, d, 0.2,0.3,0.2, 0.7,0.4,0.3, b, d, 0.2,0.3,0.4, 0.7,0.4,0.3, b, f, 0.2,0.3,0.1, 0.4,0.4,0.5, c, f, 0.2,0.3,0.3, 0.3,0.5,0.6. From this it is clear 895
International Journal of Pure Applied Mathematics that the union of two 3-polar intuitionistic fuzzy graphs is again a 3-polar intuitionistic fuzzy graph. Definition 3.9: The join, of two m-polar intuitionistic fuzzy graphs,,,, of the graphs,, respectively is defined as follows: (i) if (ii) if (iii) if where is the set of all edges joining the nodes of assuming that. (i) 0 0 if. Proposition 3.10: The join, of two m-polar intuitionistic fuzzy graphs,,,, of the graphs,, is an m-polar intuitionistic fuzzy graph of. Proof. Follows from the definition. Proposition 3.11: Let,, be crisp graphs let. Let,, be m-polar intuitionistic fuzzy subsets of,, respectively. Then,,, is an m-polar intuitionistic fuzzy graph of if only if,,,, are m-polar intuitionistic fuzzy graphs of respectively. Proof: Suppose is an m-polar intuitionistic fuzzy graph of. Let. Then,, for 1,2,3 Let. Then, for 1,2,3, 0 0. This shows that,, is an m-polar intuitionistic fuzzy graph of. Similarly, we can show that,, is an m-polar intuitionistic fuzzy graph of. The converse follows from proposition 3.8. Proposition 3.12: Let,, be crisp graphs let. Let,, be m-polar intuitionistic fuzzy subsets of,, respectively. Then,,, is an m-polar intuitionistic fuzzy graph of if only if,,,, are m-polar intuitionistic fuzzy graphs of respectively Proof. Follows from propositions 3.10 3.11. Applications Common fuzzy graphs are 1-polar fuzzy graphs. These are very much used in cluster analysis solving fuzzy intersection equations, data-base theory so on. Bipolar fuzzy graphs are further developed have been used in social networks, engineering, computer science database theory, expert systems, neural detworks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, medical diagnosis so on. m-polar intuitionistic fuzzy graphs (m>2) are very useful when analyzing real world situation involving multipolar attributes with membership values non-membership values. Moreover, the results of m-polar intuitionstic fuzzy graphs can be applicable in various areas of engineering, computer science, artificial intelligence, neural networks, social networks, so on. References [1] J. Chen, S. Li, S. Ma, X. Wang, m-polar fuzzy sets: an extension of bipolar fuzzy sets. Hindwai Publishing Corporation, Sci. World J. (2014) 8. 896
International Journal of Pure Applied Mathematics Article Id:416530, http://dx.doi.org/10.1155/2014/4165 [2] "G. Ghorai, M. Pal, On some operations density of m-polar fuzzy graphs,pac. Sci. Rev. A Nat. Sci. Eng. 17 (1) (2015) 14e22. " [3] G. Ghorai, M. Pal, A study on m-polar fuzzy planar graphs, Int. J. Comput. Sci.Math. 7 (3) (2016) 283e292. [4] G. Ghorai, M. Pal, Faces dual of m-polar fuzzy planar graphs, J. Intell. Fuzzy Syst. (2016), http://dx.doi.org/10.3233/jifs-16433. [5] F. Harary, Graph Theory, third ed., Addision- Wesely, Reading, MA, 1972. [6] "A. Kauffman, Introduction a la theorie des sousemsembles 503 flous, Masson et Cie 1, 1973. " [7] McAllister, Fuzzy intersection graphs, Comput. Math. Appl. 10 (1988) 871e886. [8] J.N. Mordeson, P.S. Nair, Fuzzy Graphs Hypergraphs, Physica Verlag, 2000. [9] "A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K.S. Fu, M. Shimura (Eds.), Fuzzy Sets Their Applications, Academic Press, New York, 1975, pp. 77e95. " [10] S. Samanta, M. Pal, Fuzzy tolerance graphs, Int. J. Latest Trends Math. 1 (2)(2011) 57e67. [11] S. Samanta, M. Pal, Fuzzy threshold graphs, CIIT Int. J. Fuzzy Syst. 3 (12) (2011)360e364. [12] S. Samanta, M. Pal, Bipolar fuzzy hypergraphs, Int. J. Fuzzy Log. Syst. 2 (1)(2012) 17e28. [13] S. Samanta, M. Pal, Irregular bipolar fuzzy graphs, Int. J. Appl. Fuzzy Sets 2(2012) 91e102. [14] S. Samanta, M. Pal, Fuzzy k-competition graphs p-competitions fuzzy graphs, Fuzzy Inf. Eng. 5 (2) (2013) 191e204. [15] S. Samanta, M. Akram, M. Pal, m-step fuzzy competition graphs, J. Appl. Math.Comput. 47 (1) (2015) 461e47 [16] S. Samanta, M. Pal, Fuzzy planar graphs, IEEE Trans. Fuzzy Syst. 23 (6) (2015)1936e1942. [17] L.A. Zadeh, Fuzzy sets, Inf. Control (1965) 338e353. [18] L.A. Zadeh, Similarity relations fuzzy ordering, Inf. Sci. 3 (1971) 177e200. [19] L.A. Zadeh, Is there a need for fuzzy logical, Inf. Sci. 178 (2008) 2751e2779. [20] W.R. Zhang, Bipolar fuzzy sets relations: a computational framework for cognitive modeling multiagent decision analysis, in: Proceedings of IEEE Conference, 1994, pp. 305e309. [21] W.R. Zhang, Bipolar fuzzy sets, in: Proceedings of Fuzzy-IEEE, 1998, pp.835e840. [22] D. Ezhilmaran, K. Sankar, Morphism of bipolar intuitionistic fuzzy graphs, Journal of Discrete Mathematical Sciences & Cryptography, Vol.18(2015), No. 5, pp. 605-621. [23] Ganesh Ghorai, Madhumangal Pal, Some properties of m-polar fuzzy graphs, Pacific Science Review A: National Science Engineering, 18(2016)38-46. [24] K.T. Atanassov, Intuitionistic Fuzzy Sets: Theory Applications, Studies in Fuzziness Soft Computing, Physica-Verl., Heidelberg, New York, 1999. [25] K.-M. Lee, Bipolar-valued fuzzy sets their basic operations, in:proceedings of the International Conference, Bangkok, Thail, 2000, pp. 307 317. 897
898