N.Sathyaseelan, Dr.E.Chandrasekaran

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1 SELF WEAK COMPLEMENTARY FUZZY GRAPHS N.Sathyaseelan, Dr.E.Chandrasekaran (Assistant Professor in Mathematics, T.K Government Arts College, Vriddhachalam ) (Associate Professor in Mathematics, Presidency College, Chennai. Tamilnadu, India.) ABSTRACT In this paper, the order, size and degree of the nodes of the isomorphic fuzzy graphs are discussed. Isomorphism between fuzzy graphs is proved to be an equivalence relation. Weak Isomorphism between fuzzy graphs is proved the partial order relation. Some properties of self complementary and self weak complementary fuzzy graphs are discussed. Keywords - Fuzzy relation, equivalence relation, weak isomorphism, Self complementary fuzzy graphs, Self weak complementary fuzzy graphs. I. INTRODUCTION The concept of fuzzy relations which has a widespread application in pattern recognition was introduced by Zadeh in his classical paper in P. Bhattacharya in [1] showed that a fuzzy graph can be associated with a fuzzy group in a natural way as an automorphism group. K.R.Bhutani in [2] introduced the concept of weak isomorphism and isomorphism between fuzzy graphs. II.PRELIMINARIES A fuzzy graph with S as the underlying set is a pair G : (σ,μ ) where σ : S [0,1] is a fuzzy subset, μ : S S [0,1] is a fuzzy relation on the fuzzy subset σ, such that μ (x, y) σ (x) σ (y) for all x,y S. Throughout this paper S is taken as a finite set.sup p(σ)={u / σ(u) > 0},andsup p(μ )={(u,v) /μ (u,v) > 0}. For the definitions that follow let G : (σ,μ ) and G ' : (σ',μ' ), be the fuzzy graphs with underlying sets S and S' respectively. Definition :2.1[4] Given a fuzzy graph G : (σ,μ ) with the underlying set S, the order of G is defined and denoted as p = (x) and size of G is defined and denoted as q= x, y) Definition :2.2 A fuzzy graph G : (σ,µ) is connected if µ (x, y) > 0 x,y σ *. Where σ * = sup (σ) = { x S / σ(x) > 0 }. Definition :2.3.[3] The complement of a fuzzy graph G : (σ,µ) is also a fuzzy graph is denoted as G c :(σ, µ c ) or G : (σ, ) where σ c = σ and µ c (x,y) = σ(x) σ(y) - µ(x,y) x,y S. III.ISOMORPHISM IN FUZZY GRAPHS Definition: 3.1[2] A homomorphism of fuzzy graph h : G G' is a map h : S S' which satisfies, σ(x) σ'[ h(x) ] and µ(x,y) µ'[ h(x), h(y) ] Definition: 3.2[2] A isomorphism h : G G' is a map h : S S' which is bijective that satisfies, σ(x) = σ' [ h(x) ] µ(x,y) = G' [ h(x), h(y) ] Isomorphism between fuzzy graphs denote as G G'. Remark : A weak isomorphism preserves the weights of the nodes but not necessarily the weights of the edges. 2. An isomorphism preserves the weights of the edges and the weights of the nodes. 3. An endomorphism of a fuzzy graph G: (σ, µ ) is a homomorphism of G to itself. 4. An automorphism of a fuzzy graph G: (σ, µ ) is an isomorphism of G to itself 5. When the two fuzzy graphs G & G' are same the weak isomorphism between them becomes an isomorphism. In crisp graph, when two graphs are isomorphic they are of same size and order. The following theorem is analogous to this. Theorem :3.4 For any two isomorphic fuzzy graph their order and size are same. Let G: (σ, µ) and G': (σ', µ') are any two fuzzy graph with the underlying set S and S' respectively. If h: G G' is an isomorphism between the fuzzy graphs G and G' then it satisfies, µ (x) = σ'[ h(x) ] µ(x, y) = µ'[ h(x), h(y) ] (i).order(g)= = Order( G') 1627 P a g e

2 (ii).size(g)= = σ(x)=σ'[h(x)] = Size(G') (1) µ(x,y)=µ'[h(x),h(y)] (2) Therefore for any two isomorphic fuzzy graph their µ (x, y) = sup { µ k (x, y) / k = 1, 2, 3, } order and size are same. Corollary:3.5 Converse of the result need not be true. Example: 3.6 Consider the fuzzy graph G and G' with the underlying sets S = {a,b,c,d} and S' = {a',b',c',d'} σ(x) = 1, µ(a,b) = 0.25, µ(b,c) = 0.5, µ(c,d) = 0.25, (x) = 1, µ'(a',b') = 0.25, µ(b',c') = 0.5, µ(c',d') = (refer fig. 3.1) From this two graphs, Order (G) = p = = (a)+ (b)+ (c)+ (d) = = 4 = Order(G') = sup { x i-1, x i ) /x = x 0,x 1,x 2,.. x k-1,x k =y S, k = 1, 2, } From equation (2), (3) = sup{ x i-1 ), h(x i )] / x = x 0, x 1, x 2, x k-1, x k = y = sup{ µ' k [h(x), h(y)] / k=1, 2, } µ (x,y)=µ [h(x),h(y)] Therefore G is isomorphic to S, k = 1, 2, } G' µ (x, y) = µ' [h(x), h(y)] Size(G) = q = = µ(a,b)+µ(b,c)+µ(c,d) = = 1 =Size(G') Theorem: 3.7 If G and G are isomorphic fuzzy graphs then the degree of their nodes are preserved. Proof Let h : S S be an isomorphism of G onto G. by the definition (3.2), µ(x,y) = µ' (h(x),h(y)) d(u) = = = d(h(u)) Therefore for any two isomorphic graphs their degrees of vertex are same. corollary:3.8 But the converse of the result need not be true. Theorem: 3.9 If G is isomorphic to G' then G is connected iff G' is also connected. Proof Assume that G is isomorphic to G'. There exist a bijective map h : S S' be an isomorphism of G onto G' such that h(x) = x',, Satisfying G is connected iff µ (x, y) > 0 iff µ' [h(x), h(y)] > 0 [by equation (3)] iff G is connected. Hence the proof. Theorem: 3.10 Isomorphism between fuzzy graphs is an equivalence relation. Let G : (σ,µ), G': (σ',μ ') and G'' : (σ'',µ'') be the fuzzy graph with the underlying set S, S' and S'' respectively. Isomorphism between fuzzy graphs is an equivalence relation if it satisfies Reflexive, Symmetric and Transitive. Reflexive Consider the identity map h : S S such that h(x)=x Since the fuzzy graphs are isomorphic then h is a bijective map, σ(x) = σ[ h(x) ] µ(x,y) = µ[ h(x), h(y) ] Hence h is an isomorphism of the fuzzy graph to itself. Therefore G is isomorphic to itself. Symmetric Let h : S S' be an isomorphism of G onto G then h is a bijective map such that 1628 P a g e

3 h(x) = x', Example:4.3(referFig4.1) (4) Satisfying σ(x) = σ' [ h(x) ] The above two fuzzy graphs are of order 3 but they are not weak isomorphic. (5) µ(x,y) = µ' [ h(x), h(y) ] Example : 4.4 (6) Let G : (σ,μ ) and G' : (σ',μ' ), be the fuzzy As h is bijective, by equation (4) h -1 (x') = x, ' Using equation (5), σ[h -1 (x')] = σ' (x') (7) µ[h -1 (x'), h -1 (y')] = µ' (x',y') ' (8) Hence we get one-to-one, onto map h -1 : S' S, which is an isomorphism from G' to G. i.e G G' G' G. Transitive Let h : S S' and g : S' S'' be an isomorphism of the fuzzy graphs G onto G' and G' onto G'' respectively. Thereforeg h : S'' S such that (g h)x g(h(x)) Since h is a bijective map that satisfies h(x) = x',. σ(x) = σ'[h(x)]andµ(x,y) = µ' [ h(x), h(y) ] i.e., σ(x) = σ' (x') and µ(x,y) = µ' (x',y') (9) Since g is a bijective map that satisfies g(x') = x'', x' S'. σ'(x') = σ'' [ g(x') ] and µ'(x',y')=µ''[g(x'),g(y')] x' S' (10) From (4), (9) and (10) σ(x) = σ' (x') = σ'' [g(x')] x' S' = σ'' [g (h(x))] σ(x) = σ'' [g h(x)] µ(x,y) = µ' (x',y') = µ'' [g(x'), g(y')] = µ'' [g(h(x)), g(h(y))] µ(x,y) = µ'' [(g h)x, (g h)y]. Therefore the transitive relation g h is an isomorphism between G and G''. Hence isomorphism between fuzzy graphs is an equivalence relation. IV. WEAK ISOMORPHISM IN FUZZY GRAPHS Definition: 4.1[2] A weak isomorphisam h : G G' is a map h : S S' which is a bijective homomorphism that satisfies, σ(x) = σ' [ h(x) ] µ(x,y) µ' [ h(x), h(y) ] Remark : 4.2 If the fuzzy graphs are weak isomorphic then their order are same. But the fuzzy graphs of same order need not be weak isomorphic. graphs with underlying sets S = {a,b.c} and S' = {a',b,c'} where σ : S [0,1], μ:s S [0,1], σ' : S' [0,1], μ':s' S' [0,1] are defined as σ(a) = 1/2, σ(b) = 1/4, σ(c) = 1/3 ; µ(a,b) = 1/5, µ(b,c) = 1/5 µ(a,c) =1/4; σ' (a') =1/2, σ' (b') =1/4, σ' (c') = 1/3 ; µ'(a',b') = 1/4, µ' (b',c') = 1/4 µ' (a',c') =1/4 Definition h : S S' as h(a) = a', h(b) = b', h(c) = c' this h is a bijective mapping σ(x) = σ' [ h(x) ] µ(x,y) µ' [ h(x), h(y) ] (refer Fig.4.2) Theorem:4.5 Weak isomorphism between fuzzy graphs satisfies the partial order relation. Proof Let G : (σ,µ), G' : (σ',µ') and G'' : (σ'',µ'') be the fuzzy graph with the underlying set S,S' and S'' respectively. Weak isomorphism between fuzzy graphs is partial order relation if it satisfies Reflexive, Anti-Symmetric and Transitive. Reflexive Consider the identity map h : S S such that h(x)=x Since the fuzzy graphs are isomorphic then h is a bijective map, σ(x) = σ[ h(x) ] µ(x,y) = µ[ h(x), h(y) ] Hence h is a weak isomorphism of the fuzzy graph to itself. Therefore G is weak isomorphic to itself. Anti Symmetric Let h be a weak isomorphism between S and S and g be a weak isomorphism between S' and S. That is h : S S' is a bijective map, h(x) = x', σ(x) = σ' [ h(x) ] (11) µ(x,y) µ' [ h(x), h(y) ] (12) g : S' S is a bijective map, g(x') = x, x' S' σ' (x') = σ[ g(x') ] ' (13) µ' (x',y') µ[ g(x'), g(y') ] ' (14) 1629 P a g e

4 The inequalities (12) and (14) hold good on the That is, G c is isomorphic to G c ' finite sets S and S' only when G and G' have the Hence G and G' are isomorphic then G c and G c ' are same number of edges and the corresponding edges also isomorphic. have same weight. Hence G and G' are identical. Sufficient Part: Transitive Assume that G c and G c ' are isomorphic then we Let h : S S' and g : S' S'' be a weak isomorphism have to prove G and G' are isomorphic fuzzy of the fuzzy graphs G onto G' and G' onto G'' graphs. respectively. By our assumption there exist a bijective map Therefore g h:s'' S such that (g h)x =g(h(x)) g : S S', σ(x) = σ' [ g(x) ] As h is a weak isomorphism, h(x) = x', (20) µ c (x,y) = µ c ' [ g(x), g(y) ] σ(x) = σ' [ h(x) ] (21) µ(x,y) µ' [ h(x), h(y) ] By the definition (2.3), (15) µ c (x,y) = σ(x) σ(y) - µ(x,y) x,y S As g is a weak isomorphism, g(x') = x'', x' S' (22) µ c ' [g(x),g(y)] = σ'[g(x)] σ' [ g(y) ]- µ' [ g(x), g(y) σ'(x') = σ'' [ g(x') ] ] µ' (x',y') µ'' [ g(x'), g(y') ] x' S'' From (20),µ c '[g(x),g(y)] =σ(x) σ(y) - µ' [ g(x), (16) From (15) and (16) g(y) ] σ(x) = σ[h(x)] = σ' (x') = σ''[g(x')] x' S' From (22), µ c (x,y) = σ(x) σ(y) - µ' [ g(x), g(y) ] = σ'' [g (h(x))] σ(x) = σ'' [g h(x)] µ(x,y) µ[h(x), h(y)] = µ' (x',y') = µ'' [g(x'), g(y')] = µ'' [g(h(x)), g(h(y))] µ(x,y) µ'' [(g h)x, (g h)y] Therefore the transitive relation g h is a weak isomorphism between G and G''. Hence weak isomorphism between fuzzy graphs is a partial order relation. Theorem :4.6 Any two fuzzy graphs are isomorphic if and only if their complements are isomorphic. Necessary Part: Let G : (σ,µ) and G': (σ',µ') be the two given fuzzy graphs. As G is isomorphic to G', there exist a bijective map h : S S' such that h(x) = x',, σ(x) = σ' [ h(x) ] (17) µ(x,y) = µ' [ h(x), h(y) ] (18) By definition (2.3), µ c (x,y) = σ(x) σ(y) - µ(x,y) x,y S (19) Using (17), µ c (x,y) = σ' [h(x)] σ' [h(y)] µ'[h(x),h(y)] x,y S Using (18), µ c (x,y) = µ c ' [h(x),h(y)] x,y S That is, µ' [ g(x), g(y) ] = σ(x) σ(y) - µ c (x,y) From (22), µ' [ g(x), g(y) ] = µ(x,y) x,y S (23) Hence from equation (20) and (23) g : S S' is an isomorphism between G and G'. Hence G c and G c ' are isomorphic then G and G' are also isomorphic. Theorem :4.7 If there is a weak isomorphism between G and G' then there is a weak isomorphism between G c and G c '. where G : (σ,µ), G': (σ',µ') are any two fuzzy graphs with the underlying set S and S'. Assume that G is a weak isomorphism between G and G'. There exist a bijective map h : S S' such that h(x) = x',, σ(x) = σ' [ h(x) ] (24) µ(x,y) µ' [ h(x), h(y) ] (25) As h -1 : S' S is also bijective for x' S' such that h -1 (x') = x. From (24), σ [h -1 (x')] = σ' (x') ' (26) By the definition of (2.3), µ c (x,y) = σ(x) σ(y) - µ(x,y) x,y S (27)Using (24) and (25) in (27) we get, 1630 P a g e

5 µ c [h -1 (x'), h -1 (y')] σ' [h(x)] σ' [h(y)] - µ' Remark:5.5 As a consequence of the above theorm [h(x),h(y)] x',y' S' σ' (x') σ' (y') - µ' if G is a self complementary fuzzy graph, then (x',y') Size of = q= (27)] µ c '(x',y') x',y' S'[by using That is, µ c ' (x',y') µ c [h -1 (x'), h -1 (y')] (28) Thus h -1 : S' S is a bijective map, which is a weak isomorphismbetweeng c andg c 'by(26)and(28). V.SELF COMPLEMENTARY FUZZY GRAPHS Definition: 5.1 A fuzzy graph G is self complementary if G G'. Where G' is the complement of fuzzy grph G. Example:5.2 (referfig: 5.1) Example: 5.3(referFig: 5.2) Theorem:5.4 G : (σ, µ) be a self complementary fuzzy graph, then (x,y) = x,y S. Let G : (σ, µ) be a self complementary fuzzy graph. Then there exists an isomorphism h : S S such that, h(x) = x, This theorem is not sufficient. Even though G is not isomorphic to G', G satisfies the equality (x,y) = x,y S. VI.SELF WEAK COMPLEMENTARY FUZZY GRAPHS Definition:6.1 A fuzzy graph G is self weak complementary if G is weak isomorphic with G c. Where G c is the complement of fuzzy graph G. σ(x) = σ[h(x)], µ(x, y) µ c [h(x), h(y)], x, y S Remark:6.2 The fuzzy graph G, fig5.2 give in example 5.3 is not a self weak complementary fuzzy graph. Example:6.3(refer Fig. 6.1) Theorem:6.4 G : (σ, µ) be a self weak complementary fuzzy graph, then (x,y) x,y S σ c [h(x)] = σ[h(x)] = σ(x) (29) µ c [h(x), h(y)] = µ[h(x), h(y)] = µ(x,y) x,y S (30) we have, µ c [h(x), h(y)] = σ c [h(x)] σc[h(y)] - µ[h(x), h(y)] ie., µ(x,y) = σ(x) σ(y) - µ[h(x), h(y)] (29)+( 30) (x,y) + [h(x),h(y)]= Using equation (2) 2 (x,y) = (x,y) =. x,y S Hence G : (σ, µ) be a self complementary fuzzy graph. Then (x,y)= x,y S. Let G : (σ, µ) be a self weak complementary fuzzy graph. Then there exists a weak isomorphism h : S S such that, h(x) = x, σ (x) = σ[h(x)] (31) µ(x, y) µ c [h(x), h(y)] x, y S (32) Now by definition(2.3), we have, µ c [h(x), h(y)] = σ[h(x)] σ[h(y)] - µ[h(x), h(y)] (33) from (32) and (33), µ(x,y) σ[h(x)] σ[h(y)] - µ[h(x), h(y)] using (31) in this above equation, µ(x,y) σ(x) σ(y) - µ[h(x), h(y)] µ(x,y) + µ[h(x), h(y)] σ(x) σ(y) Taking summation on both sides, we have, (x,y) + [h(x), h(y)] (x) σ(y)] (Since S is a finite set) 1631 P a g e

6 Therefore we have, b(1) 0.5 c(1) a (1) [h(x),h(y)] (x) σ(y)][since h(x) = b (1) x] Hence [h(x), h(y)] (x) σ(y)]in S. Theorem:6.5 If µ(x,y) x,y S then G is self weak complementary.whereg:(σ,µ) is a fuzzy graph. Fig.4.1 G : (σ,µ) G : (σ,µ ) Figure:3.1 Graphs of same order and size a (1) 0.25 b(0.75) d (0.25) 0.5 c (1) Given G : (σ,µ) is a fuzzy graph and µ(x, y) (34) Consider the identity map h : S S such that h(x) = x, and σ (x) = σ[h(x)] (35) Now by definition(2.3), we have, µ c (x, y) = σ(x) σ(y) - µ(x, y) (36) Using (34) in (36), µ c (x,y) σ(x) σ(y)- [ by equation (34)] µ c (x, y) µ c (x, y) µ(x, y) [ by equation (35)] That is, µ(x, y) µ c (x,y)=µ c [h(x),h(y)][ since h(x) = x] Therefore, µ(x, y) µ c [h(x), h(y)] (37) From equation (34) and (35) G is weak isomorphic with G c. Therefore G is a self weak complementary fuzzy graph. Remark:6.6 When G is a self weak complementary graph then order(g) =order (G c ) and size(g) size(g c ) But the converse of the above is not true. As in example, we have order(g) =order (G c ) = 2.8 and size(g)= 1.6 and size(g c ) = 1.6. but G is not a self weak complementary fuzzy graph. VII. FIGURES Fig.3.1 c (1) a(1) d(1) d (1) d(0.25) c(1) a (1) 0.25 b (0.75 G : (σ,µ) G : (σ,µ ) Fig:4.1 Graphs of same order and size but not weak isomorphic The above two fuzzy graphs are of order 3 but they are not weak isomorphic. Fig.4.2 c(1/3) c'(1/3) 1/4 1/5 1/3 1/4 a(1/2) 1/5 b(1/4) a' (1/2) 1/4 b' (1/4) Fig.5.1 G :(σ,µ) G' :(σ',µ') Figure: 4.2 Weak isomorphism a(0.6) a(0.6) b(0.8) 0.4 c(1) b(0.8) 0.4 c(1) G :(σ,µ) G c :(σ,µ c ) Fig: 5.1self complementary fuzzy graph In this example G G c P a g e

7 Fig.5.2 c(0.6) 0.3 b(1) c(0.6) 0.3 b(1) d(0.4) 0.4 a(0.8) d(0.4) a(0.8) (i) G :(σ,µ) (ii) G c (σ,µ c ) Fig:5.2 Fuzzy graphs are not Self complement In this example G G c. [5] Mordeson, J.N. and P.S. Nair Fuzzy Graphs and Fuzzy Hypergraphs Physica Verlag, Heidelberg 1998; Second Edition [6] Nagoorgani, A., and Chandrasekaran, V.T., Domination in fuzzy graph,adv. in Fuzzy sets & Systems 1(1) (2006), [7] Nagoorgani. A., and Basheer Ahamed, M., Strong and WeakDomination in Fuzzy Graphs, East Asian Mathematical Journal, Vol.23,No.1, June 30, (2007) pp 1-8. Fig.6.1 d(0.4) 0.3 c(0.6) d (0.4) 0.3 c(0.6) a(0.8) 0.3 b(1) a (0.8) 0.5 b(1) (i) G :(σ,µ) (ii) G c :(σ,µ c Fig :6.1 self weak complementary fuzzy graph. Conclusion In this paper isomorphism between fuzzy graphs is proved to be an equivalence relation and weak isomorphism is proved. To be a partial order relation. A necessary and then a sufficient condition for a fuzzy graph to be self weak complementary are studied. The results discussed may be used to study about various fuzzy graph invariants. REFERENCES 1] Bhattacharya, P, Some Remarks on fuzzy graphs, Pattern Recognition Letter 6: , [2] Bhutani, K.R., On Automorphism of Fuzzy graphs, Pattern Recognition Letter 9: ,1989 [3] Sunitha, M.S., and Vijayakumar,.A, Complement of fuzzy graph,indian J,.pure appl, Math.,33(9); September [4] Somasundaram. A., and Somasundaram, S., Domination in fuzzygraphs, Pattern Recognition Letter 19: (1998) P a g e