Irregular Bipolar Fuzzy Graphs

Transcription

1 Inernational Journal of pplications of Fuzzy Sets (ISSN 4-40) Vol ( 0), 9-0 Irregular ipolar Fuzzy Graphs Sovan Samanta ssamantavu@gmailcom Madhumangal Pal mmpalvu@gmailcom Department of pplied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore - 7 0, India bstract In this paper, we define irregular bipolar fuzzy graphs and its various classifications Size of regular bipolar fuzzy graphs is derived The relation between highly and neighbourly irregular bipolar fuzzy graphs are established Some basic theorems related to the stated graphs have also been presented Keywords: ipolar fuzzy graphs, irregular bipolar fuzzy graphs, totally irregular bipolar fuzzy graphs Introduction In 965, Zadeh [5] introduced the notion of fuzzy subset of a set as a method of presenting uncertainty The fuzzy systems have been used with success in last years, in problems that involve the approximate reasoning It has become a vast research area in different disciplines including medical and life sciences, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multiagent systems, pattern recognition, robotics, computer networks, expert systems, decision making, automata theory etc In 975, Rosenfeld [0] introduced the concept of fuzzy graphs In 994, Zhang [6] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets ipolar fuzzy sets are extension of fuzzy sets whose range of membership degree is [,] In bipolar fuzzy set, membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree within (0,] of an element indicates that the element somewhat satisfies the property, and the membership degree within [,0) of an element indicates the 9

2 element somewhat satisfies the implicit counter property For example, sweetness of foods is a bipolar fuzzy set If sweetness of foods has been given as positive membership values then bitterness foods is for negative membership values Other tastes like salty, sour, pungent (eg chili), etc are irrelevant to the corresponding property So these foods are taken as zero membership values In 0, kram [] introduced the concept of bipolar fuzzy graphs and defined different operations on it ipolar fuzzy graph theory is now growing and expanding its applications The theoretical developments in this area is discussed here Review of literature fter Rosenfeld [0], fuzzy graph theory is increased with a large number of branches Samanta and Pal introduced fuzzy tolerance graphs [], fuzzy threshold graphs [], fuzzy competition graphs [3] and bipolar fuzzy hypergraphs [4] Mathew and Sunitha [6] described the types of arcs in a fuzzy graph Nagoorgani and Malarvizhi [] established the isomorphism properties of strong fuzzy graphs Nagoorgani and Radha [] defined regular fuzzy graphs Nagoorgani and Vadivel [3] showed relations between the parameters of independent domination and irredundance in fuzzy graphs Nagoorgani and Vijayalaakshmi [4] defined insentive arc in domination of fuzzy graph Nair and Cheng [7] defined cliques and fuzzy cliques in fuzzy graphs Nair [8] established the definition of perfect and precisely perfect fuzzy graphs Natarajan [9] et al showed strong (weak) domination in fuzzy graphs kram [] defined different operations on bipolar fuzzy graphs He also shown that the automorphism property of bipolar fuzzy graph Strong bipolar fuzzy graphs were also introduced here He also introduced regular bipolar fuzzy graphs [] Preliminaries fuzzy set on a set X is characterized by a mapping m : X [0,], called the membership function fuzzy set is denoted as ( X, m) fuzzy graph [0] ξ ( V, σ, µ ) is a non-empty set V together with a pair of functions σ : V [0,] and µ : V V [0,] such that for all u, v V, µ σ ( u) σ ( (here x y denotes the minimum of x and y ) Partial fuzzy subgraph ξ ( V, τ, ν ) of ξ is such that τ ( σ ( for all v V and µ ν for all u, Fuzzy subgraph [8] ξ ( P, σ, µ ) of ξ is such that P V, ' σ ( u) σ ( u) for all u P, µ ( u, µ for all u, v P fuzzy graph is complete [] if µ σ ( u) σ ( for all u, The degree of vertex u is d( u) µ ξ minimum degree of ξ is δ ( ξ ) { d( u) u V} The maximum degree of ξ is ( ξ ) { d( u) u V} The total degree [] of a vertex u V is td( u) d( u) σ ( u) fuzzy graph ξ ( V, σ, µ ) is said to be regular [] if d ( k, a positive real number, for all v V If each vertex of ξ has same total degree k, then ξ is said to be a totally regular fuzzy graph fuzzy graph is said to be irregular [6], if there is a vertex which is adjacent to vertices with distinct degrees fuzzy graph is said to be neighbourly irregular [6], if every two adjacent vertices of the graph have different degrees fuzzy graph is said to be totally irregular, if there is a vertex which is adjacent to vertices with distinct total degrees If every two adjacent vertices The 9

3 have distinct total degrees of a fuzzy graph then it is called neighbourly total irregular [6] fuzzy graph is called highly irregular [6] if every vertex of G is adjacent to vertices with distinct degrees The complement [8] of fuzzy graph ξ ( V, σ, µ ) is the fuzzy graph ξ ( V, σ, µ ) where σ ( u) σ ( u) for all u V and 0, if µ > 0, µ ( u, σ ( u) σ (, otherwise Let X be a nonempty set bipolar fuzzy set [6] on X is an object having the form { m ( x)) x X}, where m : X [0,] and m : X [,0] are mappings If m ( x) 0 and m (x) 0, it is the situation that x is regarded as having only positive satisfaction for If m (x) 0 and m ( x) 0, it is the situation that x does not satisfy the property of but somewhat satisfies the counter property of It is possible for an element x to be such that m ( x) 0 and m ( x) 0 when membership function of the property overlaps that of its counter property over some portion of X For the sake of simplicity, we shall use the symbol (, m m ) for the bipolar fuzzy set { m ( x)) x X} For every two bipolar fuzzy sets ( m, m ) and ( m, m ) on X, ( ( x) ( min( m ( x)), max( m ( x))) ( ( x) ( max( m ( x)), min( m ( x))) bipolar fuzzy graph [] with an underlying set V is defined to be the pair G (, where ( m, m ) is a bipolar fuzzy set on V and (, m m ) is a bipolar fuzzy set on E V V such that m min{ m( x), m( } and m max{ m( x), m( } for all E Here is called bipolar fuzzy vertex set of V, the bipolar fuzzy edge set of E bipolar fuzzy graph G (, is said to be strong if m min( m ( ) m max( m ( ) The complement [] of a strong bipolar fuzzy graph G is G (, where ( m, m ) is a bipolar fuzzy set on V and (, m m ) is a bipolar fuzzy set on E V V such that () V V, () m ( x) m( x) and m ( x) m( x) for all x V, (3) 0, if m > 0, m m ( x) m (, otherwise and m 0, m ( x) m (, if m < 0, otherwise Definition [] Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The graph G is called complete bipolar fuzzy graph if m u, min{ m ( u), m ( )} and m max{ m ( u), m ( } for all u, ( v 93

4 Definition [] Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively d u) k, d ( u) ( k, k ) If ( k for all u V, k,k are two real numbers, then the graph is called -regular bipolar fuzzy graph Definition 3 [] Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The total degree of a vertex u V is denoted by td (u) and defined as td( u) ( td ( u), td ( u)) where td ( u) m m ( u), td ( u) m m ( u) If all the vertices of a bipolar fuzzy graph are of total degree, then the graph is said to be totally regular bipolar fuzzy graph 3 Some definitions related to bipolar fuzzy graphs Degree of a vertex of a bipolar fuzzy graph is defined below Definition 4 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The positive degree of a vertex u G is d u) m Similarly negative degree of a ( vertex u G is d ( u) m The degree of a vertex u is d( u) ( d ( u), d ( u)) Example We consider a bipolar fuzzy graph shown in Figure 0 Here d ( v ) , d ( v ) ( 03) ( 0) 05 So d ( v ) (09, 05) Similarly d ( v ) (09, 07) and d ( v 3 ) (, 08) Order and size of a bipolar fuzzy graph is an important term in bipolar fuzzy graph theory They are defined below Definition 5 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The order of G is denoted by O (G) and defined as O( G) ( O ( G), O ( G)) where O ( G) m ( u) and O ( G) m ( u) u V u V 94

5 Definition 6 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The size of G is defined by S( G) ( S ( G), S ( G)) where S ( G) m, S ( G) m, u v, u v Example In Figure, a bipolar fuzzy graph is shown Here O ( G) (9, 4) and S ( G) (4, ) Figure : Degrees of the vertices of a bipolar fuzzy graph Definition 7 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The underlying crisp graph of G is the crisp graph G ( V, E ) where V { v m ( > 0 or m ( ) < 0} and E { m > 0 or m ) < 0} v v Definition 8 bipolar fuzzy graph is said to be connected if its underlying crisp graph is connected Example 3 crisp underlying graph is shown in Figure It is also an example of connected bipolar fuzzy graph 95

6 Figure : Example of connected regular bipolar fuzzy graph ' Theorem Let G be a regular bipolar fuzzy graph where induced crisp graph G is an even cycle Then G is regular bipolar fuzzy graph if and only if either m and m are constant functions or alternate edges have same positive membership values and negative membership values Proof Let G (, be a regular bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively and underlying crisp graph G of G be an even cycle If either m, m are constant functions or alternate edges have same positive and negative membership values, then G is a regular bipolar fuzzy graph Conversely, suppose G is a ( k, k ) -regular bipolar fuzzy graph Let e, e,, en be the edges of G in order s in the theorem 3, c, if i is odd, m ( e i ) k c, if i is even c, if i is odd, m ( e i ) k c, if i is even If c k c, then m is constant If c k c, then alternate edges have same positive and negative membership values Similarly for m Hence the results Theorem The size of a ( k, k ) -regular bipolar fuzzy graph is ( pk, pk ) where p V Proof Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The size of G is S( G) ( m, m ) We have u v d( [ m, m ] S( G) So S( G) d( ie S( G) ( k, k) ( G ) ( pk, pk u v This gives S ) Hence the result 96

7 Theorem 3 If G is ( k, k ) -totally regular bipolar fuzzy graph, then S ( G) O( G) ( pk, pk ) where p V Proof Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and V V respectively Since G is a ( k, k ) -totally regular fuzzy graph So k td ( d ( m ( and k td ( d ( m ( for all v V Therefore k d ( m ( and k d ( m ( pk S ( G ) and pk S ( G) So pk pk ( S ( G) S ( G)) O ( G) O ( G) Hence S ( G) O( G) ( pk, pk ) 4 Irregular bipolar fuzzy graphs Irregular bipolar fuzzy graphs are as important as regular bipolar fuzzy graphs We now define it Definition 9 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively G is said to be irregular bipolar fuzzy graph if there exists a vertex which is adjacent to a vertex with distinct degrees Example 4 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively, where V { v, v, v, } d ( v ) (08, ), d ( v ) (08, ), d ( v 3 ) (, 4), 3 v4 d ( v 4 ) (04, 04) Here d( d( v3) So this graph is an example of irregular bipolar fuzzy graph, shown in Figure 3 Figure 3: Example of irregular bipolar fuzzy graph 97

8 Neighbourly irregular bipolar fuzzy graph is a special case of irregular bipolar fuzzy graph Definition 0 Let G be a connected bipolar fuzzy graph Then G is called neighbourly irregular bipolar fuzzy graph if for every two adjacent vertices of G have distinct degrees Definition Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively G is said to be totally irregular bipolar fuzzy graph if there exists a vertex which is adjacent to a vertex with distinct total degrees Example 5 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively Let V { v (05, 04), v (04, 06), (03,0)} d ( v ) (07, ), d ( v ) (06, 3) and v3 d ( v 3 ) (05, ) Here td ( v ) (, 5) and td ( v ) (, 9) So this graph is an example of totally irregular bipolar fuzzy graph Definition Let G be a connected bipolar fuzzy graph Then G is called neighbourly totally irregular bipolar fuzzy graph if for every two adjacent vertices of G have distinct total degrees Definition 3 Let G be a connected bipolar fuzzy graph Then G is called highly irregular bipolar fuzzy graph if every vertex of G is adjacent to vertices with distinct degrees Example 6 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively where V { v, v, v3} d ( v ) (06, 05), d ( v ) (08, 04) and d ( v 3 ) (05, 04) The graph is an example of highly irregular bipolar fuzzy graph highly irregular bipolar fuzzy graph need not be neighbourly irregular bipolar fuzzy graph s for example we consider a bipolar fuzzy graph of vertices v (05, 04), v(06, 05), v3(05, 04), v4(04, 04) with m v, v ) 04, m ( v, v ) 03, m v, v ) 0, m ( v, v ) 04, ( ( 3 3 ( v, v4) 0, m ( v, v4) 04 ( v3, v4) 04, m ( v3, v4) ( v) (04, 03), d( (08, ), d( v3) (06, 07), d( v4) (06, 07) ( v3) d( v4 m, m 03 So d fuzzy graph is highly irregular but not neighbourly irregular as d ) Here the bipolar Theorem 4 Let G be a bipolar fuzzy graph Then G is highly irregular bipolar fuzzy graph and neighbourly irregular bipolar fuzzy graph if and only if the degrees of all vertices of G are distinct 98

9 Proof Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and V V respectively Let V { v, v,, v } We assume that G is highly irregular and neighbourly irregular bipolar fuzzy n graphs Let the adjacent vertices of u be u, u, 3, un with degrees ( k, k ),( k3, k3 ), ( k n, k n ) respectively s G is highly and neighbourly irregular, d( u) d( u) d( u3) d( un) So it is obvious that all vertices are of distinct degrees Conversely, assume that the degrees of all vertices of G are distinct This means that every two adjacent vertices have distinct degrees and to every vertex the adjacent vertices have distinct degrees Hence G is neighbourly irregular and highly irregular fuzzy graphs Complement of a bipolar fuzzy graph is defined below Definition 4 Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively The complement of a bipolar fuzzy graph G is G (, where ( m, m ) is a bipolar fuzzy set on V and ( m, m ) is a bipolar fuzzy set on E V V such that () V V, () m ( x) m( x) and m ( x) m( x) for all x V, (3) 0, if m > 0, m m ( x) m (, otherwise m 0, m ( x) m (, if m < 0, otherwise If a bipolar fuzzy graph G is neighbourly irregular, then G is not be neighbourly irregular Let us consider a bipolar fuzzy graph whose non adjacent vertices are of same degree Then clearly, adjacent vertices of G are of same degree Hence the statement is true Theorem 5 Let G be a bipolar fuzzy graph If G is neighbourly irregular and m, m are constant functions, then G is a neighbourly total irregular bipolar fuzzy graph Proof Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and E V V respectively ssume that G is a neighbourly irregular bipolar fuzzy graph ie the degrees of every two adjacent vertices are distinct Consider two adjacent vertices u and u with distinct degrees ( k, k ) and ( k, k ) respectively lso let, m ( u) c for all u V, m ( u) c for all u V where c (0,], c [,0) are constant Therefore td u ) ( d ( u ) c, d ( u ) c ) ( k c, k ), ( c ( u) ( d ( u) c, d ( u) c) ( k c, k c td ) Clearly td u ) td( ) Therefore for any ( u 99

10 two adjacent vertices u and u with distinct degrees, its total degrees are also distinct, provided m, m are constant functions The above argument is true for every pair of adjacent vertices in G Theorem 6 Let G be a bipolar fuzzy graph If G is neighbourly total irregular and m, m are constant functions, then G is a neighbourly irregular bipolar fuzzy graph Proof Let G (, be a bipolar fuzzy graph where ( m, m ) and ( m, m ) be two bipolar fuzzy sets on a non-empty finite set V and V V respectively ssume that G is a neighbourly total irregular bipolar fuzzy graph ie the total degree of every two adjacent vertices are distinct Consider two adjacent vertices u and u with degrees ( k, k ) and ( k, k ) respectively lso assume that m ( u) c for all u V, m ( u) c for all u V where c (0,], c [,0) are constants lso td( u) td( u) We are to prove that d( u) d( u) k c k c and k c k c So k k and k k Hence the degrees of adjacent vertices of G are distinct This is true for every pair of adjacent vertices in G Hence the result 5 Conclusion In this paper we have described degree of a vertex, order, size and underlying crisp graph of a bipolar fuzzy graphs The necessary and sufficient conditions for a bipolar fuzzy graph to be the regular bipolar fuzzy graphs have been presented Size of a bipolar fuzzy graphs and relation between size and order of a bipolar fuzzy graphs have been calculated We have defined irregular bipolar fuzzy graphs, neighbourly irregular, totally and highly irregular bipolar fuzzy graphs Some relations about the defined graphs have been proved Some theoretical discussions on bipolar fuzzy graphs have been made We want to make, in near future, some algorithm and models using these results References [] M kram, ipolar fuzzy graphs, Information Sciences, doi:006/jins , 0 [] M kram, W Dudek, Regular bipolar fuzzy graphs, Neural Computing and pplications, DOI: 0007/s [3] K R hutani and attou, On M-strong fuzzy graphs, Information Sciences, 55( ), 03-09, 003 [4] K R hutani and Rosenfeld, Strong arcs in fuzzy graphs, Information Sciences, 5, 39-3,

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12 [8] P S Nair, Perfect and precisely perfect fuzzy graphs, Fuzzy Information Processing Society, 9-, 008 [9] C Natarajan and S K yyasawamy, On strong (weak) domination in fuzzy graphs, World cademy of Science, Engineering and Technology, 67, 47-49, 00 [0] Rosenfeld, Fuzzy graphs, in: L Zadeh, KS Fu, M Shimura (Eds), Fuzzy Sets and Their pplications, cademic Press, New York, 77-95, 975 [] S Samanta and M Pal, Fuzzy tolerance graphs, International Journal of Latest Trends in Mathematics, (), 57-67, 0 [] S Samanta and M Pal, Fuzzy threshold graphs, CIIT International Journal of Fuzzy Systems, 3(), , 0 [3] S Samanta and M Pal, Fuzzy k-competition graphs and p-competition fuzzy graphs, Communicated [4] S samanta and M Pal, ipolar fuzzy hypergraphs, International Journal of Fuzzy Logic Systems, (), 7 8, 0 [5] L Zadeh, Fuzzy sets, Information and Control, 8, , 965 [6] WR Zhang, ipolar fuzzy sets and relations: a computational frame work for cognitive modelling and multiagent decision analysis, Proceedings of IEEE Conf, , 994 Madhumangal Pal is a Professor of pplied Mathematics with Oceanology and Computer Programming, Vidyasagar University, India He is Editor-in-Chief of Journal of Physical Sciences and member of Editorial oard of International Journal of Computer Sciences, of Systems Engineering and Information Technology, of International Journal of Fuzzy Systems & Rough Systems, of dvanced Modeling and Optimization, Romania and of International Journal of Logic and Computation, Malaysia He is a reviewer of several international journals He has written several books on Mathematics and Computer Science His research interest includes computational graph theory, fuzzy matrices, game theory and regression analysis, parallel and genetic algorithms, etc Sovan Samanta received his Sc degree in 007 and MSc degree in 009 in pplied Mathematics from Vidyasagar University, India He is now currently a research scholar in the Department of pplied Mathematics, Vidyasagar University since 00 His research interest includes fuzzy graph theory and bipolar fuzzy graph theory 0