Bipolar Intuitionistic Fuzzy Graphs with Applications

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1 Bipolar Intuitionistic Fuzzy Graphs with Applications K. Sankar 1, D. Ezhilmaran 2 1 Department of Mathematics, C. Abdul Hakeem College of Engineering & Technology, Melvisharam, Vellore, Tamilnadu, India. 1 Research Scholar, Bharathiar University, Coimbatore Tamilnadu, India. 2 School of Advanced Sciences, VIT University, Vellore Tamilnadu, India. Abstract: - The concepts of neighbourly irregular bipolar intuitionistic fuzzy graphs, neighbourly totally irregular bipolar intuitionistic fuzzy graphs, highly irregular bipolar intuitionistic fuzzy graphs and highly totally irregular bipolar intuitionistic fuzzy graphs are introduced and investigated. A necessary and sufficient condition under which neighbourly irregular and highly irregular bipolar intuitionistic fuzzy graphs are equivalent is discussed. The notion of bipolar intuitionistic fuzzy digraphs is introduced. The bipolar intuitionistic fuzzy influence graph of a social group is also described. Keywords: - Irregular bipolar intuitionistic fuzzy graph, Neighbourly irregular bipolar intuitionistic fuzzy graphs, Highly irregular bipolar intuitionistic fuzzy graph, Bipolar intuitionistic fuzzy digraphs, Bipolar intuitionistic fuzzy influence graph. I I. INTRODUCTION n 1965, Zadeh [20] introduced the notion of a fuzzy subset of a set. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multi agent systems, pattern recognition, robotics, computer networks, expert systems, decision making and automata theory. In 1994, Zhang [24] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. A bipolar fuzzy set is an extension of Zadeh s fuzzy set theory whose membership degree range is [-1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0,1] of an element indicates that the element somewhat satisfies the property, and the membership degree [-1,0) of an element indicates that the element somewhat satisfies the implicit counter-property. In 1975, Rosenfeld [16] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffmann [11] in The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analougs of several graph theoretical concepts. Bhattacharya [8] gave some remarks on fuzzy graphs. The complement of a fuzzy graph was defined by Mordeson and Nair [13]. Recently, the bipolar fuzzy graphs have been discussed in [1 3]. In 2015, D.Ezhilmaran and K.Sankar[26] have defined bipolar intuitionistic fuzzy graphs. In this paper, we introduce the concepts of neighbourly irregular bipolar intuitionistic fuzzy graphs, neighbourly totally irregular bipolar intuitionistic fuzzy graphs, highly irregular bipolar intuitionistic fuzzy graphs and highly totally irregular bipolar intuitionistic fuzzy graphs. We prove a necessary and sufficient condition under which neighbourly irregular and highly irregular bipolar intuitionistic fuzzy graphs are equivalent. We introduce the notion of bipolar intuitionistic fuzzy digraphs. We also describe the bipolar intuitionistic fuzzy influence graph of a social group. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [4 7,9,10,13 9,22,23,25]. II. PRELIMINARIES In this section, we first review some definitions of undirected graphs that are necessary for this paper. Definition 2.1[27] Recall that a graph is an ordered pair G* = (V,E), where V is the set of vertices of G* and E is the set of edges of G*. Two vertices x and 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 and no more than one edge between any two different vertices. Definition 2.2[27] A subgraph of a graph G* = (V,E) is a graph H=(W,F) where W and F. Definition 2.3[27] The complementary graph of a simple graph has the same vertices as. Two vertices are adjacent in if and only if they are not adjacent in. Definition 2.4[27] Consider the cartesian product ( ) of graphs and. Then and *( )( ) + *( )( ) + Definition 2.5[20] A fuzzy subset on a set X is a map, -. A map, - 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 2.6[12] Page 44

2 Let X be a non empty set. A bipolar fuzzy set B in X is an object having the form {( ( ) ( )) } where, - and, - are mappings. Definition 2.7[28] Let X be a non empty set. An intuitionistic fuzzy set B = {( ( ) ( )) } Where, - and, - are mapping such that ( ) ( ) III. BIPOLAR INTUITIONISTIC FUZZY GRAPHS Definition 3.1[26] Let X be a non empty set. A bipolar intuitionistic fuzzy set B = {( ( ) ( ) ( ) ( )) } where, -, -, -, - are the mappings such that ( ) ( ), ( ) ( ) We use the positive membership degree ( ) to denote the satisfaction degree of an element x to the property corressponding to a bipolar intuitionistic fuzzy set B and the negative membership degree ( ) to denote the satisfaction degree of an element x to some implicit counter property corressponding to a bipolar intuitionistic fuzzy set. Similarly we use the positive nonmembership degree ( ) to denote te satisfaction degree of an element x to the property corresponding to a bipolar intuitionistic fuzzy set and the negative nonmembership degree ( ) to denote the satisfaction degree of an element x to some implicit counter property crossponding to a bipolar intuitionistic fuzzy set. If ( ) ( ) and ( ) ( ) it is the situation that x regarded as having only the positive membership property of a bipolar intuitionistic fuzzy set. If ( ) ( ) and ( ) ( ) it is the situation that x regarded as having only the negative membership property of a bipolar intuitionistic fuzzy set. ( ) ( ) and ( ) ( ) it is the situation that x regarded as having only the positive nonmembership property of a bipolar intuitionistic fuzzy set. ( ) ( ) and ( ) ( ) it is the situation that x regarded as having only the negative nonmembership property of a bipolar intuitionistic fuzzy set. It is possible for an element x to be such that ( ) ( ) and ( ) ( ) when the membership and nonmembership function of the property overlaps with its counter properties over some portion of X. Definition 3. 2[26] For any two bipolar intuitionistic fuzzy sets. ( ) ( ) ( ) ( )/ and. ( ) ( ) ( ) ( )/ ( )( ) ( ( ) ( ) ( ) ( )) ( )( ) ( ( ) ( ) ( ) ( )) ( )( ) ( ( ) ( ) ( ) ( )) ( )( ) ( ( ) ( ) ( ) ( )) Definition 3. 3[26] Let X be a non empty set. Then we call a mapping ( ) : (, -, -, -, -) a bipolar intuitionistic fuzzy relation on X such that ( ), - ( ), - ( ), - ( ), -. Definition 3. 4[26] Let. ( ) ( ) ( ) ( )/ and. ( ) ( ) ( ) ( )/ be bipolar intuitionistic fuzzy sets on a set X. If. ( ) ( ) ( ) ( )/ is a bipolar intuitionistic fuzzy relation on if ( ) ( ) ( ), ( ) ( ) ( ),. ( ) ( ) ( ) ( )/ ( ) ( ) ( ), ( ) ( ) ( ) for all. A bipolar intuitionistic fuzzy relation A on X is called symmetric if ( ) ( ), ( ) ( ) and ( ) ( ), ( ) ( ) for all Definition 3.5[26] A bipolar intuitionistic fuzzy graph of a graph G * = (V,E) is a pair G(A,B) where A= ( ) is a bipolar intuitionistic fuzzy set in V and B = ( ) is a bipolar intuitionistic fuzzy set in V V such that ( ) ( ( ) ( )) ( ) ( ( ) ( )) ( ) ( ( ) ( )) ( ) ( ( ) ( )) and ( ) ( ) ( ) ( ) Page 45

3 Throughout this paper, G* is a crisp graph and G is a bipolar Definition 3.6 Let G be a bipolar The neighbourhood of a vertex x in G is defined by ( ) ( ( ) ( ) ( ) ( )) where ( ) { ( ) ( ( ) ( ))}, where ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ). If there is a vertex which is adjacent to vertices with distinct closed neighbourhood degrees, then G is called a totally irregular bipolar Example ( ) { ( ) ( ( ) ( ))}, ( ) { ( ) ( ( ) ( ))}, ( ) { ( ) ( ( ) ( ))} Definition 3.7 Let G be a bipolar The neighbourhood degree of a vertes x in G is defined by ( ) ( ( ) ( ) ( ) ( )) where ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( ) ( ). Notice that ( ), ( ), ( ), ( ) for and ( ), ( ), ( ), ( ) for. Definition 3.8 Let G be a bipolar intuitionistic fuzzy graph on G*. If there is a vertex which is adjacent to vertices with distinct neighbourhood degrees, then G is called an irregular bipolar Example 3.9 * + and { } deg(v 1 )=(2.0,-1.0,1.5,-2.3),deg(v 2 )=(2.0,-1.0,1.5,-2.3), deg(v 3 )=(2.0,-1.0,1.5,-2.3) deg(v 4 )=(2.7,-1.3,2.0,-3.1) and deg(v 5 )=(1.0,-0.4,0.7,-1.2). It is clear from calculations that G is totally irregular bipolar Definition 3.12 A connected bipolar intuitionistic fuzzy graph G is said to be a neighbourly irregular bipolar intuitionistic fuzzy graph if every two adjacent vertices of G have distinct open neighbourhood degree. Example 3.13 ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ). It is clear that G is an irregular bipolar Definition 3.10 Let G be a bipolar The closed neighbourhood degree of a vertex x is defined by ( ) ( ( ) ( ) ( ) ( )), deg(v 1 )=(1.2,-0.7,0.5,-1.2), deg(v 2 )=(0.9,-1.0,1.2,-0.3), deg(v 3 )=(1.2,-0.7,0.5,-1.2) deg(v 4 )=(0.9,-1.0,1.2,-0.3). It is clear from calculations that G is neighbourly irregular bipolar Page 46

4 Definition 3.14 A connected bipolar intuitionistic fuzzy graph G is said to be a neighbourly totally irregular bipolar intuitionistic fuzzy graph if every two adjacent vertices of G have distinct closed neighbourhood degree. Example 3.15 Consider a vertex which is adjacent to and with distinct neighbourhood degrees. But deg(v 2 )= deg(v 3 ). So G is highly irregular bipolar intuitionistic fuzzy graph but it is not a neighbourly irregular bipolar intuitionistic fuzzy graph. Remark 2 A neighbourly irregular bipolar intuitionistic fuzzy graph may not be a highly irregular bipolar intuitionistic fuzzy graph. Example 3.18 deg(v 1 )=(1.8,-1.1,1.3,-1.4), deg(v 2 )=(1.4,-1.3,1.4,-1.0), deg(v 3 )=(1.5,-1.3,0.9,-1.3) deg(v 4 )=(1.6,-1.4,1.5,-0.8). It is clear from calculations that G is neighbourly totally irregular bipolar Definition 3.16 Let G be a connected bipolar intuitionistic fuzzy graph. G is called a highly irregular bipolar intuitionistic fuzzy graph if every vertex of G is adjacent to vertices with distinct neighbourhood degrees. Remark 1 A highly irregular bipolar intuitionistic fuzzy graph may not be a neighbourly irregular bipolar intuitionistic fuzzy graph. There is no relation between highly irregular bipolar intuitionistic fuzzy graph and neighbourly irregular bipolar We explain this concept with the following examples. Example 3.17 By routine computations, we have deg(v 1 )=(0.5,-0.7,1.0,-1.6), deg(v 2 )=(1.0,-1.0,1.0,-1.8), deg(v 3 )=(1.0,-1.0,1.0,-1.8) deg(v 4 )=(0.8,-0.6,0.7,-1.1)and deg(v 5 )=(1.2,-1.4,1.3,-2.5). By routine computations, we have deg(v 1 )=(1.0,-0.7,0.9,-0.3), deg(v 2 )=(1.1,-0.7,1.1,-0.7), deg(v 3 )=(1.0,-0.7,0.9,-0.3) and deg(v 4 )=(1.1,-0.7,1.1,-0.7). We see that two adjacent vertices have distinct neighbourhood degrees. But consider a vertex which is adjacent to the vertex and has same degree, that is, deg(v 1 )= deg(v 3 ). Hence G is neighbourly irregular bipolar intuitionistic fuzzy graph but not a highly irregular bipolar Theorem 3.19 Let G be a bipolar Then G is highly irregular bipolar intuitionistic fuzzy graph and neighbourly irregular bipolar intuitionistic fuzzy graph if and only if the neighbourhood degrees of all the vertices of G are distinct. Proof Let G be a bipolar intuitionistic fuzzy graph with n vertices. Assume that G is highly irregular bipolar intuitionistic fuzzy graph and neighbourly irregular bipolar Claim: The neighbourhood degrees of all vertices of G are distinct. Let ( ) ( ) Let the adjacent vertices of be with neighbourhood degrees ( ) ( ) ( ) respectively. Then we have and since G highly irregular. Also and since G is neighbourly Page 47

5 irregular. Hence the neighbourhool degrees of all the vertices of G are distinct. Conversely, assume that the neighbourhood degrees of all the vertices of G are distinct. Claim: G is highly irregular and neighbourly irregular bipolar Let ( ) ( ) Given that and which implies that every two adjacent vertices have distinct neighbourhood degrees and to every vertex, the adjacent vertices have distinct neighbourhood degrees. Theorem 3.20 A bipolar intuitionistic fuzzy graph G of G*, where G* is a cycle with vertices 3 is heighbourly irregular and highly irregular bipolar intuitionistic fuzzy graph if and only if the positive membership, negative membership, positive nonmembership, negative non-membership values of the vertices between every pair of vertices are all distinct. Proof Assume that positive membership, negative membership, positive non-membership, negative nonmembership values of the vertices are all distinct. Claim: G is highly irregular and neighbourly irregular bipolar Let.Given that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) which implies that ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) ( ) ( ). That is, ( ) ( ) ( ). Hence G is a neighbourly irregular and highly irregular bipolar intuitionistic fuzzy graph. Conversely, assume that G is neighbourly and highly irregular bipolar Claim: Positive membership, negative membership, positive non-membership and negative non-membership values of the vertices are all distinct. Let ( ) ( ) Suppose that positive membership, negative membership, positive nonmembership and negative non-membership values of two vertices are same. Let Let ( ) ( ) ( ) ( ) ( ) ( ) and ( ) ( ). Then ( ) ( ). Since G* is cycle, which is a contradiction to he fact that G is neighbourly irregular and highly irregular bipolar Hence positive membership, negative membership, positive nonmembership and negative non-membership values of the vertices are all distinct. Proposition 3.21 A neighbourly irregular bipolar intuitionistic fuzzy graph may not be a neighbourly totally irregular bipolar Example.3.22 By routine compuatations, we have deg(v 1 )=(0.7,-0.9,0.5,- 0.7), deg(v 2 )=(0.8,-1.1,0.6,-0.9), deg(v 3 )=(0.7,-0.9,0.5,-0.7) and deg(v 4 )=(0.8,-1.1,0.6,-0.9) and deg, -=, -, deg, -=, -, deg, -=, - and deg, -=, -. We see that ( ) ( ). Hence G is neighbourly irregular bipolar intuitionistic fuzzy graph but not a neighbourly totally irregular bipolar Remark 3 A neighbourly totally irregular bipolar intuitionistic fuzzy graph may not be a neighbourly irregular bipolar Example 3.23 By routine computations, we have deg, -=[1.2,-1.5,0.9,- 1.2], deg, -=[1.3,-1.4,1.0,-1.1], deg, -=[1.4,-1.3,1.1,-1.0] and deg, -=[1.5,-1.2,1.2,-0.9] but deg(v 1 )=(0.9,-0.9,0.7,-0.7), deg(v 2 )=( 0.9,-0.9,0.7,-0.7), deg(v 3 )=( 0.9,-0.9,0.7,-0.7) and deg(v 4 )=( 0.9,-0.9,0.7,-0.7). Hence G is neighbourly totally irregular bipolar intuitionistic fuzzy graph but not a neighbourly irregular bipolar Proposition 3.24 Let G be a bipolar If G is neighbourly irregular bipolar intuitionistic fuzzy graph and ( ) is a constant function, then G is a neighbourly totally irregular bipolar Proof Page 48

6 Assume that G is neighbourly irregular bipolar That is neighbourhood degrees of every two adjacent vertices are distinct. Let, where and are adjacent vertices with distinct neighbourhood degrees ( ) and ( ) respectively. That is deg(v 1 )= ( ) and deg(v 2 )= ( ) where,, and. Let us assume that ( ( ) ( ) ( ) ( )). ( ) ( ) ( ) ( )/ ( ) where and are constant and, -,, -. Therefore,, - ( ) ( ) and, - ( ) ( ),, - ( ) ( ) and, - ( ) ( ). Claim:, - ( ),, - ( ),, - ( ) and, - ( ). Suppose that, - ( ),, - ( ),, - ( ) and, - ( ). Consider, - ( ) which is a contradiction to and, - ( ) which is a contradiction to. Similarly and. Therefore, - ( ),, - ( ),, - ( ) and, - ( ). Hence G is neighbourly totally irregular bipolar Remark 4 If G is neighbourly irregular bipolar intuitionistic fuzzy graph, then the bipolar intuitionistic fuzzy subgraph ( ) of G may not be highly irregular bipolar Example 3.25 * + and * +. Consider ( ) such that * + and * + and ( ) such that 1) For G : by routine computations, we have deg(v 1 )=(1.2,-0.6,1.0,-0.8), deg(v 2 )=(2.5,-1.3,2.1,- 1.8), deg(v 3 )=(1.3,-0.4,1.1,-0.6), deg(v 4 )=(1.8,- 0.7,1.5,-1.0) and deg(v 5 )=(1.9,-0.6,1.6,-0.9). It is easy to see that G is neighbourly irregular bipolar 2) For : by routine computations, we have deg(v 1 )=(1.2,-0.6,1.0,-0.8), deg(v 2 )=(1.9,-0.8,1.6,- 1.1), deg(v 4 )=(1.2,-0.6,1.0,-0.8), deg(v 5 )=(1.9,- 0.6,1.6,-0.9). It is easy to see that is neighbourly irregular bipolar 3) For : by routine computations, we have deg(v 2 )=(1.3,-0.3,1.1,-0.5), deg(v 3 )=(1.3,-0.4,1.1,- 0.6), deg(v 4 )=(1.2,-0.3,1.0,-0.5). It is easy to see that is neighbourly irregular bipolar intuitionistic fuzzy graph. 4) For : by routine computations, we have deg(v 2 )=(1.2,-0.6,1.0,-0.8), deg(v 2 )=(1.3,-0.4,1.1,- 0.6), deg(v 4 )=(1.2,-0.6,1.0,-0.8), deg(v 5 )=(0.6,- 0.4,0.5,-0.5). It is easy to see that and are adjacent vertices with same neighbourhood degree in. Hence is not a beighbourly irregular bipolar intuitionistic fuzzy graph but g is neighbourly irregular bipolar Remark 5 If G is totally irregular bipolar intuitionistic fuzzy graph, then bipolar intuitionistic fuzzy subgraph ( ) need not be totally irregular bipolar Example 3.26 G Page 49

7 . Consider ( ) such that * + and * + 1) For G : by routine computations, we have deg, - =(1.4,-2.0,1.1,-1.7), deg, - =(1.8,-2.8,1.4,- 2.4), deg, -=(1.3,-2.1,1.0,-1.8) and deg, -=(1.8,- 2.8,1.4,-2.4). Here there is a vertex which is adjacent to and where deg, -, -, -. 2) For H : by routine computation, we have deg, - =(1.4,-2.0,1.0,-1.7), deg, - =(1.4,-2.0,1.0,- 1.7), deg, -=(0.5,-0.7,0.4,-0.6). Here, it is easy to see that there is no vertex whose adjacent vertices having different neighbourhood degrees. Hence H is not a totally irregular bipolar intuitionistic fuzzy graph but G is totally irregular bipolar intuitionistic fuzzy graph. IV. BIPOLAR INTUITIONISTIC FUZZY DIGRAPHS A directed graph ( or digraph ) is a graph whose edges have directions and are called arcs ( edges ). Arrows on the arcs are used to encode the directional information: an arc from vertex x to vertex y indicates that one may move from x to y but not from y to x. Let ( ) and ( ) be two digraphs. The Cartesian product of and gives a digraph ( ) with ( ) and *( ) ( ) + *( ) ( ) +. In this section, we will write to mean, and if, we say x and y are adjacent such that x is a starting node and y is an ending node. Throughout this paper, we denote D* a crisp simple digraph and D a bipolar intuitionistic fuzzy digraph. Definition 4.1 A bipolar intuitioistic fuzzy digraph of a digraph D* is a pair D=(A,B) where ( ) is a bipolar intuitionistic fuzzy set in V and ( ) is a bipolar intuitionistic fuzzy relation on E such that ( ) ( ( ) ( )), ( ) ( ( ) ( )), ( ) ( ( ) ( )), ( ) ( ( ) ( )) for all. We note B need not to be symmetric. Example 4.2 Consider a graph D*=(V,E) such that * + and * +. Let A be a bipolar ituitionistic fuzzy set of V and let B be a bipolar intuitionistic relation of defined by It is easy to see that, the bipolar intuitionistic fuzzy digraph graph D represented by the following adjacency matrix Definition 4.3 Let ( ) and ( ) be bipolar intuitionistic fuzzy digraphs of. Then is called a bipolar intuitionistic fuzzy subdigraph of if ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ) for all. We write. Let ( ) and ( ) be bipolar intuitionistic fuzzy subsets of and and let ( ) and ( ) be bipolar intuitionistic fuzzy subsets of and respectively. The Cartesian product of two bipolar intuitionistic fuzzy digraphs and of the digraphs and is denoted by ( ) and is defined as follows i) ( )( ) ( ( ) ( )) ( )( ) ( ( ) ( )) ( )( ) ( ( ) ( )) ( ) ( )( ) ( ( ) ( )) for all Page 50

8 ii) ( )( )( ) ( ( ) ( )) for all, iii) ( )( )( ) ( ( ) ( )) for all,. Proposition 4.4 If and are the bipolar intuitionistic fuzzy digraphs then is also a bipolar intuitionistic fuzzy digraph. Definition 4.5 A bipolar intuitionistic fuzzy digraph ( ) is called a strong bipolar intuitionistic fuzzy digraph if ( ) ( ( ) ( )), ( ) ( ( ) ( )), ( ) ( ( ) ( )) and ( ) ( ( ) ( )) for all. Proposition 4.6 If and are the strong bipolar intuitionistic fuzzy digraphs then is a strong bipolar intuitionistic fuzzy digraph. V. APPLICATION OF BIPOLAR INTUITIONISTIC FUZZY DIGRAPHS Graph models find wide application in many areas of mathematics, computer science, the natural and social sciences. Often these models need to incorporate more structure than simply the adjacencies between vertices. In studies of group behavior, it is observed that certain people can influence thinking of others. A directed graph, called an influence graph, can be used to model this behavior. Each person of a group is represented by a vertex. There is a directed edge from vertex to vertex when the person represented by a vertex influence the person represented by vertex. This graph does not contain loops and it does not contain multiple directed edges. In any social group all the persons can never be members of the group always. Any person can be removed from the group at any time if his activity is against the group. Any person can leave the group at any time if he feels the group is not suitable for him. Though a person removed or withdrawn himself from the group there will be some change in the powers and influence over the other members of the group. Each person of the group is represented by a vertex. Every vertex have four values the first value represents the positive power of the person in the group which means how much power he posses to control the group the second value represents the negative power which arise when some persons in the group are against him. The third and fourth values represents the positive and negative powers of the person in the group when he become nonmember (removed or withdrawn himself from the group) of the group. Every directed edge have four values the first and second values represents the positive and negative influence by the first person over the second person when the first person is member of the group. The third and fourth values represents the positive and negative influence by the first person over the second person when the first person is a nonmember of the group In the figure-14 the person represented by posses 20% of positive power 40% of negative power in the group when he is member of the group and 30% of positive and 10% negative power respectively over the group when he is nonmember of the group. The person represented by influence the person, 50% positively and 10% negatively and influence the person, 60% positively and 40% negatively when is a member and a nonmember of the group respectively. REFERENCES [1]. M. Akram, Bipolar fuzzy graphs, Information Sciences 181 (2011) [2]. M. Akram, W.A. Dudek, Regular bipolar fuzzy graphs, Neural Computing & Applications 1 (2012) [3]. M. Akram, M.G. Karunambigai, Metric in bipolar fuzzy graphs, World Applied Sciences Journal 14 (2011) [4]. M. Akram, W.A. Dudek, Interval-valued fuzzy graphs, Computers &Mathematics with Applications 61 (2011) [5]. M. Akram, W.A. Dudek, Intuitionistic fuzzy hypergraphs with applications, Information Sciences (2012). [6]. M. Akram, A.B. Saeid, K.P. Shum, B.L. Meng, Bipolar fuzzy K- algebras, International Journal of Fuzzy System 10 (3) (2010) [7]. Y. Alavi, G. Chartrand, F.R.K. Chung, P. Erdös, R.L. Graham, O.R. Oellermann, Highly irregular graphs, Journal of Graph Theory 11 (2) (1987) [8]. P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letter 6(1987) [9]. S.G. Bhragsam, S.K. Ayyaswamy, Neighbourly irregular graphs, Indian Journal of Pure and Applied Mathematics 35 (3) (2004) [10]. A.N. Gani, S.R. Latha, On irregular fuzzy graphs, Applied Mathematical Sciences 6 (2012) [11]. A. Kauffman, Introduction a la Theorie des Sous-emsembles Flous, Paris:Masson et Cie Editeurs, [12]. K.-M. Lee, Bipolar-valued fuzzy sets and their basic operations, in: Proceedings of the International Conference, Bangkok, Thailand, 2000, pp [13]. J.N. Mordeson, P.S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, second ed., Physica Verlag, Heidelberg 1998, [14]. S. Mathew, M.S. Sunitha, Node connectivity and arc connectivity of a fuzzy graph, Information Sciences 180 (4) (2010) Page 51

9 [15]. F. Riaz, K.M. Ali, Applications of graph theory in computer science, in: 2011 Third International Conference Computational Intelligence, Communication Systems and Networks (CICSyN), 2011, pp [16]. A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K.S. Fu, M. Shimura (Eds.), Fuzzy Sets and their Applications, Academic Press, New York, 1975, pp [17]. S.G. Shirinivas, S. Vetrivel, N.M. Elango, Applications of graph theory in computer science an overview, International Journal of Engineering Science and Technology 2 (9) (2010) [18]. M.S. Sunitha, A. Vijayakumar, Complement of a fuzzy graph, Indian Journal of Pure and Applied Mathematics 33 (9) (2002) [19]. H.-L. Yang, S.-G. Li, Z.-L. Guo, C.-H. Ma, Transformation of bipolar fuzzy rough set models, Knowledge-Based Systems 27 (2012) [20]. L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) [21]. L.A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences 3 (2)(1971) [22]. L.A. Zadeh, Is there a need for fuzzy logic?, Information Sciences 178 (2008) [23]. [W.-R. Zhang, Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, in: Proceedings of IEEE Conference, 1994, pp [24]. W.-R. Zhang, Bipolar fuzzy sets, in: Proceedings of FUZZ-IEEE, 1998, pp [25]. W.-R. Zhang, YinYang Bipolar Relativity, IGI Global, [26]. D. Ezhilmaran, K. Sankar, Morphism of bipolar intuitionistic fuzzy graphs, Journal of Discrete Mathematical Sciences & Cryptography, Vol.18(2015), No. 5, pp [27]. F. Harary, Graph Theory, third ed., Addison-Wesley, Reading, MA, [28]. K.T. Atanassov, Intuitionistic Fuzzy Fets: Theory and Applications, Studies in Fuzziness and Soft Computing, Physica- Verl., Heidelberg, New York, Page 52