Ji2416271633

223 views

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
223
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
2
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Ji2416271633

  1. 1. N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.1627-1633 SELF WEAK COMPLEMENTARY FUZZY GRAPHS N.Sathyaseelan ∗ , Dr.E.Chandrasekaran∗∗ ∗(Assistant Professor in Mathematics, T.K Government Arts College, Vriddhachalam – 606 001.) ∗∗(Associate Professor in Mathematics, Presidency College, Chennai. Tamilnadu, India.)ABSTRACT  In this paper, the order, size and degree of the G : (σ,  ) where σc = σ and µc(x,y) = σ(x) σ(y) -nodes of the isomorphic fuzzy graphs are µ(x,y) x,y S.discussed. Isomorphism between fuzzy graphs isproved to be an equivalence relation. Weak III.ISOMORPHISM IN FUZZY GRAPHSIsomorphism between fuzzy graphs is proved Definition: 3.1[2]the partial order relation. Some properties of A homomorphism of fuzzy graph h :self complementary and self weak G→G is a map h : S→S which satisfies,complementary fuzzy graphs are discussed. σ(x) ≤ σ[ h(x) ] x S andKeywords - Fuzzy relation, equivalence relation,weak isomorphism, Self complementary fuzzy µ(x ,y) ≤ µ[ h(x), h(y) ] x Sgraphs, Self weak complementary fuzzy graphs. Definition: 3.2[2] A isomorphism h : G→G is a map h :I. INTRODUCTION S→S which is bijective that satisfies, The concept of fuzzy relations which has σ(x) = σ [ h(x) ] x Sa widespread application in pattern recognition was µ(x,y) = G [ h(x), h(y) ] x Sintroduced by Zadeh in his classical paper in 1965.P. Bhattacharya in [1] showed that a fuzzy graph Isomorphism between fuzzy graphscan be associated with a fuzzy group in a natural denote as G G.way as an automorphism group. K.R.Bhutani in Remark : 3.3[2] introduced the concept of weak isomorphism 1. A weak isomorphism preserves the weights ofand isomorphism between fuzzy graphs. the nodes but not necessarily the weights of the edges.II.PRELIMINARIES 2. An isomorphism preserves the weights of the A fuzzy graph with S as the underlying set edges and the weights of the nodes.is a pair G : (σ ,μ ) where σ : S →[0,1] is a fuzzy 3. An endomorphism of a fuzzy graph G: (σ , µ ) issubset, μ : S × S →[0,1] is a fuzzy relation on a homomorphism of G to itself.the fuzzy subset σ , such that μ (x, y) ≤ σ (x)∧ σ (y) 4. An automorphism of a fuzzy graph G: (σ , µ ) isfor all x,y ∈S . an isomorphism of G to itselfThroughout this paper S is taken as a finite set.sup 5. When the two fuzzy graphs G & G are same thep(σ)={u / σ(u) > 0},andsup p(μ )={(u,v) /μ (u,v) > weak isomorphism between them becomes an0}. For the definitions that follow let G : (σ ,μ ) isomorphism .and G : (σ ,μ ), be the fuzzy graphs withunderlying sets S and S respectively. In crisp graph, when two graphs are isomorphicDefinition :2.1[4] they are of same size and order. The following theorem is analogous to this.Given a fuzzy graph G : (σ ,μ ) with the underlying Theorem :3.4set S, the order of G is defined and denoted as p For any two isomorphic fuzzy graph their= (x) and size of G is defined and denoted as order and size are same.q= x, y)Definition :2.2 Proof:Let G: (σ, µ) and G: (σ, µ) are any two A fuzzy graph G : (σ,µ) is connected if fuzzy graph with the underlying set S and Sµ∞(x, y) > 0 x,y σ*. Where σ* = sup (σ) = { x respectively. If h: G→ G is an isomorphism S / σ(x) > 0 }. between the fuzzy graphs G and G then it satisfies, µ (x) = σ[ h(x) ] x SDefinition :2.3.[3] µ(x, y) = µ[ h(x), h(y) ] x S The complement of a fuzzy graph G : (σ,µ) isalso a fuzzy graph is denoted as Gc :(σ , µc) or (i).Order(G)= = Order( G) 1627 | P a g e
  2. 2. N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.1627-1633(ii).Size(G)= = σ(x)=σ[h(x)] x S = Size(G) (1) µ(x,y)=µ[h(x),h(y)] x S (2)Therefore for any two isomorphic fuzzy graph their µ∞(x, y) = sup { µk(x, y) / k = 1, 2, 3,… }order and size are same. = sup { xi-1, xi) /x = x0,x1,x2,.. xk-1,xk =yCorollary:3.5 Converse of the result need not be S, k = 1, 2,…}true. From equation (2),Example: 3.6 = sup{ xi-1), h(xi)] / x = x0, x1, x2,… Consider the fuzzy graph G and G withthe underlying sets S = {a,b,c,d} and S ={a,b,c,d} σ(x) = 1 x S, µ(a,b) = 0.25, xk-1, xk = y S, k = 1, 2,…}µ(b,c) = 0.5, µ(c,d) = 0.25, (x) = 1 x S, = sup{ µk [h(x), h(y)] / k=1, 2,…}µ(a,b) = 0.25, µ(b,c) = 0.5, µ(c,d) = 0.25. (referfig. 3.1) µ∞(x,y)=µ’∞[h(x),h(y)] (3) From this two graphs, Therefore G is isomorphic toOrder (G) = p = = (a)+ (b)+ (c)+ (d)=1+1+1+1 = 4 = Order(G) G µ∞(x, y) = µ∞[h(x), h(y)] x SSize(G) = q = = G is connected iff µ∞(x, y) > 0 iff µ∞[h(x), h(y)] >µ(a,b)+µ(b,c)+µ(c,d) = .25+.25+.5= 1 =Size(G) 0 [by equation (3)]Theorem: 3.7 iff G’ is connected. Hence the proof. If G and G’ are isomorphic fuzzy graphs Theorem: 3.10then the degree of their nodes are preserved. Isomorphism between fuzzy graphs is anProof equivalence relation.Let h : S→S’ be an isomorphism of G onto G. by Proof:the definition (3.2), Let G : (σ,µ), G: (σ,μ ) and G : (σ,µ) be∑µ(x,y) = ∑µ (h(x),h(y)) x S the fuzzy graph with the underlying set S, S and S respectively. Isomorphism between fuzzy graphs d(u) = = = is an equivalence relation if it satisfies Reflexive , Symmetric and Transitive.d(h(u)) ReflexiveTherefore for any two isomorphic graphs theirdegrees of vertex are same. Consider the identity map h : S→S such that h(x)=x x Scorollary:3.8 But the converse of the result neednot be true. Since the fuzzy graphs are isomorphic then h is a bijective map satisfying, σ(x) = σ[ h(x) ] x STheorem: 3.9 If G is isomorphic to G then G is µ(x,y) = µ[ h(x), h(y) ] x Sconnected iff G is also connected. Hence h is an isomorphism of the fuzzy graph toProof itself. Therefore G is isomorphic to itself. Symmetric Assume that G is isomorphic to G. Let h : S→S be an isomorphism of G ontoThere exist a bijective map h : S→S be an G’ then h is a bijective map such thatisomorphism of G onto G such that h(x) = x, x S,Satisfying 1628 | P a g e
  3. 3. N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.1627-1633h(x) = x, x S Example:4.3(referFig4.1)(4)Satisfying σ(x) = σ [ h(x) ] x S The above two fuzzy graphs are of order 3 but they are not weak isomorphic.(5)µ(x,y) = µ [ h(x), h(y) ] x S Example : 4.4(6) Let G : (σ ,μ ) and G : (σ ,μ ), be the fuzzyAs h is bijective, by equation (4) h-1(x) = x, x S graphs with underlying sets S = {a,b.c} and S =Using equation (5), σ[h-1(x)] = σ (x) x S (7) {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,µ[h-1(x), h-1(y)] = µ (x,y) x S 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 ;(8) µ(a,b) = 1/4, µ (b,c) = 1/4 µ (a,c) =1/4 Hence we get one-to-one, onto map Definition h : S →S as h(a) = a, h(b) = b , h(c) =h-1 : S→S, which is an isomorphism from G to G. c this h is a bijective mapping satisfying i.e G G G G. σ(x) = σ [ h(x) ] x S Transitive µ(x,y) ≤ µ [ h(x), h(y) ] x S Let h : S→S and g : S→S be anisomorphism of the fuzzy graphs G onto G and G (refer Fig.4.2)onto G respectively. Theorem:4.5Thereforeg h : S→S such that (g h)x g(h(x)) Weak isomorphism between fuzzy graphs satisfiesx S the partial order relation.Since h is a bijective map that satisfies h(x) = x,x S. Proofσ(x) = σ[h(x)]andµ(x,y) = µ [ h(x), h(y) ] Let G : (σ,µ), G : (σ,µ) and G : (σ,µ) be thex S fuzzy graph with the underlying set S,S and Si.e., σ(x) = σ (x) and µ(x,y) = µ (x,y) x S respectively. Weak isomorphism between fuzzy(9) graphs is partial order relation if it satisfiesSince g is a bijective map that satisfies Reflexive , Anti-Symmetric and Transitive.g(x) = x, x S.σ(x) = σ [ g(x) ] and Reflexiveµ(x,y)=µ[g(x),g(y)] x S Consider the identity map h : S→S such that(10) h(x)=x x SFrom (4), (9) and (10)σ(x) = σ (x) = σ [g(x)] x S Since the fuzzy graphs are isomorphic then h is a bijective map satisfying,= σ [g (h(x))] x Sσ(x) = σ [g h(x)] x S σ(x) = σ[ h(x) ] x Sµ(x,y) = µ (x,y) = µ [g(x), g(y)] = µ [g(h(x)), g(h(y))] x S µ(x,y) = µ[ h(x), h(y) ] x Sµ(x,y) = µ [(g h)x, (g h)y] x S. Hence h is a weak isomorphism of the fuzzy graphTherefore the transitive relation g h is an to itself. Therefore G is weak isomorphic to itself. Anti – Symmetricisomorphism between G and G. Let h be a weak isomorphism between S and S’ andHence isomorphism between fuzzy graphs is an g be a weak isomorphism between S and S.equivalence relation. That is h : S→S is a bijective map, h(x) = x, x S satisfyingIV. WEAK ISOMORPHISM IN FUZZY σ(x) = σ [ h(x) ] x SGRAPHS (11)Definition: 4.1[2] µ(x,y) ≤ µ [ h(x), h(y) ] x S A weak isomorphisam h : G → G is a (12)map h : S →S which is a bijective homomorphism g : S→S is a bijective map, g(x) = x, x Sthat satisfies, satisfyingσ(x) = σ [ h(x) ] x S σ (x) = σ[ g(x) ] x Sµ(x,y) ≤ µ [ h(x), h(y) ] x S (13)Remark : 4.2 If the fuzzy graphs are weak µ (x,y) ≤ µ[ g(x), g(y) ] x Sisomorphic then their order are same. But the fuzzy (14)graphs of same order need not be weak isomorphic. 1629 | P a g e
  4. 4. N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.1627-1633The inequalities (12) and (14) hold good on the That is, Gc is isomorphic to Gcfinite sets S and S only when G and G have the Hence G and G are isomorphic then Gc and Gc aresame number of edges and the corresponding edges also isomorphic.have same weight. Hence G and G are identical. Sufficient Part: Transitive Assume that Gc and Gc are isomorphic then weLet h : S→S and g : S→S be a weak isomorphism have to prove G and G are isomorphic fuzzyof the fuzzy graphs G onto G and G onto G graphs.respectively. By our assumption there exist a bijective mapTherefore g h:S→S such that (g h)x =g(h(x)) g : S→S satisfying,x S σ(x) = σ [ g(x) ] x SAs h is a weak isomorphism, h(x) = x, x S (20)satisfying µc(x,y) = µc [ g(x), g(y) ] x Sσ(x) = σ [ h(x) ] (21)µ(x,y) ≤ µ [ h(x), h(y) ] x S By the definition (2.3),(15) µc(x,y) = σ(x) σ(y) - µ(x,y) x,y SAs g is a weak isomorphism, g(x) = x, x S (22)satisfying µc [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),µc[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), µc(x,y) = σ(x) σ(y) - µ [ g(x), g(y) ] = σ [g (h(x))] x S That is, µ [ g(x), g(y) ] = σ(x) σ(y) - µc(x,y)σ(x) = σ [g h(x)] x S From (22), µ [ g(x), g(y) ] = µ(x,y) x,y Sµ(x,y) ≤ µ[h(x), h(y)] = µ (x,y) = µ [g(x), g(y)] (23) = µ [g(h(x)), g(h(y))] Hence from equation (20) and (23) g : S→S is anµ(x,y) ≤ µ [(g h)x, (g h)y] x S isomorphism between G and G.Therefore the transitive relation g h is a weak Hence Gc and Gc are isomorphic then G and G are also isomorphic.isomorphism between G and G. Theorem :4.7Hence weak isomorphism between fuzzy graphs is If there is a weak isomorphism between G and Ga partial order relation. then there is a weak isomorphism between G c and Gc. where G : (σ,µ), G: (σ,µ) are any two fuzzyTheorem :4.6 graphs with the underlying set S and S.Any two fuzzy graphs are isomorphic if and only if Proof:their complements are isomorphic. Assume that G is a weak isomorphism between GProof: and G.Necessary Part: There exist a bijective map h : S→S such thatLet G : (σ,µ) and G: (σ,µ) be the two given fuzzy h(x) = x, x S satisfying,graphs.As G is isomorphic to G, there exist a bijective σ(x) = σ [ h(x) ] x Smap h : S →S such that h(x) = x, x S satisfying, (24) µ(x,y) ≤ µ [ h(x), h(y) ] x Sσ(x) = σ [ h(x) ] x S (25)(17) As h-1 : S→S is also bijective for x S such thatµ(x,y) = µ [ h(x), h(y) ] x S h-1(x) = x.(18) From (24), σ [h-1 (x)] = σ (x) x SBy definition (2.3), (26)µc(x,y) = σ(x) σ(y) - µ(x,y) x,y S By the definition of (2.3),(19) µc(x,y) = σ(x) σ(y) - µ(x,y) x,y SUsing (17), µc(x,y) = σ [h(x)] σ [h(y)] (27)Using (24) and (25) in (27) we get, –µ[h(x),h(y)] x,y SUsing (18), µc(x,y) = µc [h(x),h(y)] x,y S 1630 | P a g e
  5. 5. N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.1627-1633µc[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= c ≥ µ (x,y) x,y S[by using(27)] This theorem is not sufficient. Even though G is not isomorphic to G, G satisfies the equalityThat is, µc (x,y) ≤ µc[h-1(x), h-1 (y)] (x,y) = x,y S.(28)Thus h-1 : S→S is a bijective map, which is a weak VI.SELF WEAK COMPLEMENTARYisomorphismbetweenGcandGcby(26)and(28). FUZZY GRAPHS Definition:6.1V.SELF COMPLEMENTARY FUZZYGRAPHS A fuzzy graph G is self weak complementary if GDefinition: 5.1 is weak isomorphic with Gc. Where Gc is theA fuzzy graph G is self complementary if G ≅G. complement of fuzzy graph G.Where G is the complement of fuzzy grph G. σ(x) = σ[h(x)], x SExample:5.2 (referFig: 5.1) µ(x, y) ≤ µc[h(x), h(y)], x, y SExample: 5.3(referFig: 5.2) Remark:6.2 The fuzzy graph G, fig5.2 give inTheorem:5.4 example 5.3 is not a self weak complementary G : (σ, µ) be a self complementary fuzzy graph, fuzzy graph.then Example:6.3(refer Fig. 6.1) (x,y) = x,y S. Theorem:6.4Proof: G : (σ, µ) be a self weak complementary fuzzyLet G : (σ, µ) be a self complementary fuzzy graph. graph, then (x,y) ≤Then there exists an isomorphism h : S→Ssuch that, h(x) = x x S satisfying, x,y S σc[h(x)] = σ[h(x)] = σ(x) x S Proof: Let G : (σ, µ) be a self weak complementary fuzzy graph. Then there exists a(29) weak isomorphism h : S→S such that, h(x) = µc[h(x), h(y)] = µ[h(x), h(y)] = µ(x,y) x,y S x satisfying,(30) σ (x) = σ[h(x)] x S we have, (31)µc[h(x), h(y)] = σc [h(x)] σc[h(y)] - µ[h(x), h(y)] µ(x, y) ≤ µc[h(x), h(y)] x, y Sie., µ(x,y) = σ(x) σ(y) - µ[h(x), h(y)] (32)(29)+( 30)⟹ (x,y) + Now by definition(2.3), we have, [h(x),h(y)]= µc[h(x), h(y)] = σ[h(x)] σ[h(y)] - µ[h(x), h(y)]Using equation (2) (33) 2 (x,y) = from (32) and (33), (x,y) = . x,y S 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)]Hence G : (σ, µ) be a self complementary fuzzygraph. Then µ(x,y) + µ[h(x), h(y)] ≤ σ(x) σ(y) Taking summation on both sides, we have, (x,y)= x,y S. (x,y) + [h(x), h(y)] ≤ (x) σ(y)] (Since S is a finite set) 1631 | P a g e
  6. 6. N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.1627-1633Therefore we have, b(1) 0.5 c(1) a’(1) 0.252 [h(x),h(y)]≤ (x) σ(y)][since h(x) = b’(1)x] G : (σ,µ) G’ : (σ’,µ’)Hence [h(x), h(y)] ≤ (x) σ(y)]inS. Figure:3.1 Graphs of same order and sizeTheorem:6.5 Fig.4.1If µ(x,y) ≤ x,y S then G is selfweak complementary.WhereG:(σ,µ) is a fuzzygraph. a (1) 0.25 b(0.75) d’(0.25) 0.5Proof: c’(1)Given G : (σ,µ) is a fuzzy graph and 0.5 00 0.75 0.75 0.75µ(x, y) ≤ d(0.25) 0.125 c(1) a’(1) 0.25 b’(0.75(34) G : (σ,µ) G’ : (σ’,µ’)Consider the identity map h : S→S such thath(x) = x, x S and Fig:4.1 Graphs of same order and size but not weak isomorphicσ (x) = σ[h(x)] x S(35) The above two fuzzy graphs are of order 3 but they are not weak isomorphic.Now by definition(2.3), we have,µc(x, y) = σ(x) σ(y) - µ(x, y) Fig.4.2(36)Using (34) in (36), c(1/3) c(1/3)µc(x,y)≥σ(x) σ(y)- [ by equation(34)] 1/4 1/5 1/3 1/4µc(x, y) ≥µc(x, y) ≥ µ(x, y) x S[ by equation (35)]That is, µ(x, y)≤µ (x,y)=µc[h(x),h(y)][ since h(x) = cx]Therefore, µ(x, y) ≤ µc[h(x), h(y)] x S a(1/2) 1/5 b(1/4) a (1/2) 1/4 b (1/4)(37)From equation (34) and (35) G is weak isomorphic G :(σ,µ) G :(σ,µ)with Gc. Therefore G is a self weak complementaryfuzzy graph. Figure: 4.2 Weak isomorphismRemark:6.6 Fig.5.1When G is a self weak complementary graph then a(0.6) a(0.6)order(G) =order (Gc) and size(G) ≤ size(Gc)But the converse of the above is not true. 0.3 0.3 0.3 0.3As in example , we have order(G) =order (Gc) = 2.8and size(G)= 1.6 and size(Gc) = 1.6. but G is not aself weak complementary fuzzy graph.VII. FIGURESFig.3.1 a(1) d(1) d’(1) 0.25c’(1) b(0.8) 0.4 c(1) b(0.8) 0.4 c(1) 0.25 0.25 G :(σ,µ) Gc :(σ,µc)0.5 Fig: 5.1self complementary fuzzy graph In this example G ≅ Gc. 1632 | P a g e
  7. 7. N.Sathyaseelan, Dr.E.Chandrasekaran / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 4, July-August 2012, pp.1627-1633Fig.5.2 [5] Mordeson, J.N. and P.S. Nair Fuzzy c(0.6) 0.3 b(1) c(0.6) 0.3 Graphs and Fuzzy Hypergraphs Physicab(1) Verlag, Heidelberg 1998; Second Edition 2001. 03 0.6 0.1 0.4 0.6 [6] Nagoorgani, A., and Chandrasekaran,0.2 V.T., Domination in fuzzy graph,Adv. in 0 Fuzzy sets & Systems 1(1) (2006), 17-26. [7] Nagoorgani. A., and Basheer Ahamed, M., Strong and WeakDomination in Fuzzy Graphs, East Asian Mathematical Journal, d(0.4) 0.4 a(0.8) d(0.4) Vol.23,No.1, June 30, (2007) pp 1-8.a(0.8) (i) G :(σ,µ) (ii) Gc (σ,µc)Fig:5.2 Fuzzy graphs are not Self complementIn this example G ≇ Gc.Fig.6.1d(0.4) 0.3 c(0.6) d (0.4) 0.3 c(0.6)0.2 0.1 0.2 0.2 0.4 0.5 0.4 a(0.8) 0.3 b(1) a (0.8) 0.5 b(1) (i) G :(σ,µ) (ii) Gc :(σ,µcFig :6.1 self weak complementary fuzzy graph.Conclusion In this paper isomorphism between fuzzy graphsis proved to be an equivalence relation and weakisomorphism is proved. To be a partial orderrelation. A necessary and then a sufficientcondition for a fuzzy graph to be self weakcomplementary are studied. The results discussedmay be used to study about various fuzzy graphinvariants.REFERENCES 1] Bhattacharya, P, Some Remarks on fuzzy graphs, Pattern Recognition Letter 6: 297- 302, 1987. [2] Bhutani, K.R., On Automorphism of Fuzzy graphs, Pattern Recognition Letter 9: 159- 162,1989 [3] Sunitha, M.S., and Vijayakumar,.A, Complement of fuzzy graph,Indian J,.pure appl, Math.,33(9); 1451-1464 September 2002. [4] Somasundaram. A., and Somasundaram, S., Domination in fuzzygraphs, Pattern Recognition Letter 19: (1998) 787-791. 1633 | P a g e

×