Bi-Magic labeling of Interval valued Fuzzy Graph

Transcription

1 Advaces i Fuzzy Mathematics. ISSN X Volume 1, Number 3 (017), pp Research Idia Publicatios Bi-Magic labelig of Iterval valued Fuzzy Graph K.Ameeal Bibi 1 ad M.Devi 1, PG ad Research Departmet of Mathematics D.K.M College for Wome (Autoomous), Vellore-63001,Tamiladu, Idia. Abstract I this paper, we itroduced the cocept of Bi-Magic labelig of Iterval valued fuzzy graph. Here, we discussed the importat observatios o Bi- Magic labelig of Iterval valued fuzzy (IVF) graph. We acquired some of their properties ad also desiged some structures ad carry over them ito the operatios of iterval valued fuzzy Bi-Magic labeled graphs. We attaied eighbourhood itervals ad defied the membership values of the vertices ad edges. Further, we ivestigated Iterval valued fuzzy Bi-Magic labelig i some special graphs such as fuzzy cycle graph, fuzzy star graph ad fuzzy path graph. Keywords: Fuzzy Bi-Magic labelig, Iterval valued Fuzzy Bi-magic labelig, IVF cycle graph, IVF Star graph, IVF Path graph. AMS Mathematical Subject Classificatio: 03E7, 08A7, 05C7, 05C INTRODUCTION For more sufficiet descriptio of ucertaity, Zadeh [13] itroduced i 1975, the cocept of iterval valued fuzzy set which is a geeralizatio of traditioal fuzzy set [1,13]. It is,therefore promiet to use iterval valued fuzzy set i applicatios such as Fuzzy Automatio cotrol, Commuicatio Networks, Optimizatio theory, Medical diagostic system ad Remote sesig which are the most favourable. Nagoor Gai et al[8,9] itroduced the cocept of labelig ad magic labelig of fuzzy graphs. Oly fuzzy graphs remai scaty to solve all the problems exist i real life. The cocept of iterval valued fuzzy graph is defied by Akram ad Dudek[1], ad may other researchers, Ismayil ad Ali[5], Talebi ad Rashmaolu[11], Debath,

2 646 K.Ameeal Bibi ad M.Devi S.N.Mishra ad Aita pal[6] obtaied various properties of these graphs. Here, we itroduced the cocept of Bi-Magic labelig for iterval-valued fuzzy graphs. The graphs which are cosidered i this paper are simple, fiite, coected ad udirected.. PRELIMINARIES: Let U ad V be two o-empty sets. The is said to be a fuzzy relatio from U ito V if is a fuzzy set of UxV. A fuzzy graph G :, is a pair of fuctios : V [0,1] ad : V V [0,1 ], where for all u,v V, we have ( u, v) ( u) ( v). A graph admits fuzzy labelig if the mappig ad defied above are bijectios such that the membership value of edges ad vertices are distict ad ( u, v) ( u) ( v) for all u,v V. Defiitio.1 A fuzzy labelig graph admits Bi-magic labelig if the sum of the membership values of vertices ad edges icidet at the vertices are k1 ad k where k1 ad k are costats ad deoted by m ( G) B o A fuzzy labelig graph which admits a Bi-magic labelig is called a Fuzzy Bi-magic labelig graph. Defiitio.([1]) By a iterval valued fuzzy graph of a graph G, we mea a pair G * =(A,B) where A = [μ A, μ A + ] is a iterval valued fuzzy set o V ad B = [μ B, μ B + ] is a iterval valued fuzzy relatio o E such that μ A (xy) mi (μ A (x), μ A (y)) Defiitio.3([6]) μ A + (xy) mi (μ A + (x), μ A + (y)) for all x,y E. A graph G * =(A,B) is said to be a iterval valued fuzzy graph if μ A, μ A +, μ B, μ B + [0,1] are all distict for all vertices ad edges where μ A ad μ B are all the lower limits of the iterval membership of vertices ad edges ad μ A + ad μ B + are the upper limits of the iterval membership of vertices ad edges respectively.

3 Bi-Magic labelig of Iterval valued Fuzzy Graph 647 Defiitio.4 A iterval [μ δ, μ + δ] is said to be a δ-eighbourhood of ay membership value for ay δ satisfyig the followig coditios. (i) δ mi {μ v (v i ), μ e (e ij )} (ii) δ 1 max {μ v (v i ), μ e (e ij )} (iii) δ or d(μ(x), μ(y)) Where d(μ(x), μ(y))= μ(x) μ(y) ad μ(x), μ(y) are the membership of vertices or edges. Theorem.5([6]) Ay fuzzy graph ca be coverted ito a iterval valued fuzzy labelig graph. 3. INTERVAL VALUED FUZZY BI-MAGIC LABELING GRAPH: Defiitio 3.1 A iterval valued fuzzy labelig graph is said to be a iterval valued fuzzy Bi- Magic graph if the sum of lower membership values (ie)., μ A (x) + μ B (x, y)+μ A (y) of vertices ad edges icidet at the vertices are k1 ad k where k1 ad k are costats. Similarly, if the sum of upper membership values (ie)., μ A + (x) + μ B + (x, y)+μ A + (y) of vertices ad edges icidet at the vertices are k1 ad k where k1 ad k are costats. * The lower Bi-Magic labelig is defied as Bm 0 ( G ) ad upper Bi-Magic labelig is * defied as Bm 0 ( G ). Cosider a Cycle graph with fuzzy labeled vertices ad edges. I this graph, the lower limits are μ A (v 1 ) + μ B (v 1, v ) + μ A (v ) = = 0.11 μ A (v ) + μ B (v, v 3 )+μ A (v 3 ) = = 0.11 μ A (v 3 ) + μ B (v 3, v 1 ) + μ A (v 1 ) = = 0.14 for all v 1, v, v 3 V. Here k1=0.11 ad k=0.14 ad the upper limits are μ A + (v 1 ) + μ B + (v 1, v ) + μ A + (v ) = = 1.1 μ A + (v ) + μ B + (v, v 3 )+μ A + (v 3 ) = = 1.1 μ A + (v 3 ) + μ B + (v 3, v 1 ) + μ A + (v 1 ) = = 1.4

4 648 K.Ameeal Bibi ad M.Devi for all v 1, v, v 3 V. Here k1=1.1 ad k =1.4 Therefore the above graph admits fuzzy Bi-Magic labelig. Example 3. Theorem 3.3([6]) A fuzzy graph which admits magic labelig is called a Fuzzy magic labeled graph. Every Fuzzy magic graph ca be coverted ito iterval valued fuzzy magic graph. Result 3.4 A fuzzy graph which admits Bi-magic labelig is called a Fuzzy Bi-magic labeled graph. Every Fuzzy Bi-magic graph ca be coverted ito a iterval valued fuzzy Bimagic graph. Example 3.5

5 Bi-Magic labelig of Iterval valued Fuzzy Graph 649 I this example, we cosidered a fuzzy Bi-Magic labeled graph G with Bi-Magic values 1.1, 1.4. Now take δ=0.01, to get δ-eighbourhood itervals for G *.Thus we obtaie d the iterval valued fuzzy graph which satisfies all the coditios of Bi- Magic labelig of iterval valued fuzzy graph. Theorem 3.6 If is odd, the the cycle graph C is always a iterval valued fuzzy Bi-Magic graph. Proof: Let G be a cycle with odd umber of vertices ad v 1, v, v 3, v ad v 1 v, v v 3, v v 1 be the vertices ad edges of C. Let δ [0,1] such that oe ca choose δ 1 = 0.01 ad δ = 0.1 for lower ad upper limit respectively for 3 ad we ca choose δ 1 = ad δ = 0.01 for lower ad upper limit respectively for 4 ad the membership itervals are defied as follows: μ A (v i ) = ( 4 i)δ δ 1, 1 i 3 μ A (v i ) = ( 4 i)δ δ 1, i 1 μ A + (v i ) = ( 4 i)δ + δ 1, 1 i 3 μ A + (v i ) = ( 4 i)δ + δ 1, i 1 μ A (v i 1 ) = Max {μ A (v i )/1 i 1 } iδ for 1 i + 1 μ + A (v i 1 ) = Max {μ + A (v i )/1 i 1 } iδ for 1 i + 1 μ B (v 1, v ) = 1 Max{μ A (v i )/1 i } μ B + (v 1, v ) = 1 Max{μ A + (v i )/1 i } μ B (v i+1, v i ) = μ B (v 1, v ) iδ for 1 i 1 μ B + (v i+1, v i ) = μ B + (v 1, v ) iδ for 1 i 1

6 650 K.Ameeal Bibi ad M.Devi Here, we ivestigated the results for a iterval valued fuzzy Bi-Magic cycle for =7. Case (i) : for i is eve The i=z for ay positive iteger z For each edge v i, v i+1 Bm 0 ) = μ A (v i ) + μ B (v i, v i+1 ) + μ A (v i+1 ) = μ A (v z ) + μ B (v z, v z+1 ) + μ A (v z+1 ) = {( 4 z)δ δ 1 /1 i 3 }+1 Max{μ A (v i )/1 i }- ( z)δ + Max {μ A (v i )/1 i 1 } (z + 1)δ = 1 Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 }+( 5)δ δ 1. If i=z for ay positive iteger z, (z 1 ) Bm 0 ) = μ A (v z ) + μ B (v z, v z+1 ) + μ A (v z+1 ) = {( 4 z)δ δ 1 /i 1 }+1 Max{μ A (v i )/1 i }- ( z)δ + Max {μ A (v i )/1 i 1 } (z + 1)δ = 1 Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 }+(3 1)δ δ 1. BM 0 ) = μ + A (v z ) + μ + B (v z, v z+1 ) + μ + A (v z+1 ) = 1 Max{μ A + (v i )/1 i }+ Max {μ A + (v i )/1 i 1 }+( 5)δ + δ 1. If i=z for ay positive iteger z, (z 1 ) BM 0 ) 1)δ + δ 1. = 1 Max{μ A + (v i )/1 i }+ Max {μ A + (v i )/1 i 1 }+(3 Case (ii): for i is odd The i=z+1 for ay positive iteger z For each edge v i, v i+1 Bm 0 ) = μ A (v i ) + μ B (v i, v i+1 ) + μ A (v i+1 )

7 Bi-Magic labelig of Iterval valued Fuzzy Graph 651 = μ A (v z+1 ) + μ B (v +1z, v z+ ) + μ A (v z+ ) = Max {μ A (v i )/1 i 1 }-(z + 1)δ + 1 Max{μ A (v i )/1 i }- ( z + 4)δ + ( z)δ δ 1 = 1 Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 }+( 5)δ δ 1. Bm 0 ) = μ A (v z+1 ) + μ B (v +1z, v z+ ) + μ A (v z+ ) = {( z)δ δ 1 /i 1 }+1 Max{μ A (v i )/1 i }- ( z 8)δ + Max {μ A (v i )/1 i 1 } (z + 1)δ = 1 Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 }+(3 1)δ δ 1. BM 0 ) = 1 Max{μ A + (v i )/1 i }+ Max {μ + A (v i )/1 i 1 }+( 5)δ + δ 1. BM 0 ) = 1 Max{μ A + (v i )/1 i }+ Max {μ + A (v i )/1 i 1 }+(3 1)δ + δ 1. I Geeral, Bm ( C ) =zδ δ Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 0 Bm ( C ) =( + )δ δ Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 BM BM 0 0 ( C 0 ( C ) ) =zδ δ Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 } =( + )δ δ Max{μ A (v i )/1 i }+ Max {μ A (v i )/1 i 1 } From the above discussio, the cycle with odd umber of vertices is a Iterval valued fuzzy Bi-Magic labeled graph. } } Theorem: 3.7 For ay 4, a fuzzy labeled Star graph S1, is always a Iterval valued fuzzy Bi- Magic graph. Proof: Let S1, be the Star graph with v, u 1, u, u as vertices ad vu 1, vu,, vu as edges.

8 65 K.Ameeal Bibi ad M.Devi Let δ [0,1] such that oe ca choose δ 1 = ad δ = 0.01 for lower ad upper limits respectively for 4 ad the membership itervals are defied as follows: μ A (u i ) = [( + 1) i]δ δ 1 for i = 1,,3 μ A + (u i ) = [( + 1) i]δ + δ 1 for i = 1,,3 μ A (u i ) = {[( + 1) i] 1}δ δ 1 for 4 i μ A + (u i ) = {[( + 1) i] 1}δ + δ 1 for 4 i μ B (v, u 1 ) = Max{μ A (v), μ A (u 1 )} Mi{μ A (v), μ A (u 1 )} δ δ 1 for i = 1 μ B (v, u ) = Max{μ A (v), μ A (u )} Mi{μ A (v), μ A (u )} δ 1 for i = μ B (v, u 3 ) = Max{μ A (v), μ A (u 3 )} Mi{μ A (v), μ A (u 3 )} + δ δ 1 for i = 3 μ B (v, u i ) = Max{μ A (v), μ A (u i )} Mi{μ A (v), μ A (u i )} + 3δ δ 1 for 4 i μ B + (v, u 1 ) = Max{μ A + (v), μ A + (u 1 )} Mi{μ A + (v), μ A + (u 1 )} δ + δ 1 for i = 1 μ B + (v, u ) = Max{μ A + (v), μ A + (u )} Mi{μ A + (v), μ A + (u )} + δ 1 for i = μ B + (v, u 3 ) = Max{μ A + (v), μ A + (u 3 )} Mi{μ A + (v), μ A + (u 3 )} + δ + δ 1 for i = 3 μ B + (v, u i ) = Max{μ A + (v), μ A + (u i )} Mi{μ A + (v), μ A + (u i )} + 3δ + δ 1 for 4 i μ A (v) = μ A (u i ) ( + 1)δ 1 for 1 i

9 Bi-Magic labelig of Iterval valued Fuzzy Graph 653 μ + A (v) = μ A + (u i ) ( + 1)δ 1 for 1 i The the costats k 1 & k of the Iterval valued fuzzy Bi-Magic labelig are defied as follows: To fid k 1: Case (i) for i=1 Bm ( S ) = μ A (v) + μ B (v, u 1 ) + μ A (u 1 ) BM 0 1, = { μ A (u i ) ( + 1)δ 1 /1 i } + Max{μ A (v), μ A (u 1 )} Mi{μ A (v), μ A (u 1 )} ( + 3)δ 1 + ( 1)δ 0 ( S1, ) = μ + A (v) + μ + B (v, u 1 ) + μ + A (u 1 ) = { μ A + (u i ) ( + 1)δ 1 /1 i } + Max{μ + A (v), μ + A (u 1 )} Mi{μ + A (v), μ + A (u 1 )} ( 1)δ 1 + ( 1)δ Case (ii) for i= Bm ( S ) = μ A (v) + μ B (v, u ) + μ A (u ) BM 0 1, = { μ A (u i ) ( + 1)δ 1 /1 i } + Max{μ A (v), μ A (u )} Mi{μ A (v), μ A (u )} ( + 3)δ 1 + δ 0 ( S1, ) = μ + A (v) + μ + B (v, u ) + μ + A (u ) ={ μ A + (u i ) ( + 1)δ 1 /1 i } + Max{μ + A (v), μ + A (u )} Mi{μ + A (v), μ + A (u )} ( 1)δ 1 + δ Case (iii) for i=3 Bm ( S ) = μ A (v) + μ B (v, u 3 ) + μ A (u 3 ) 0 1, ={ μ A (u i ) ( + 1)δ 1 /1 i } + Max{μ A (v), μ A (u 3 )} Mi{μ A (v), μ A (u 3 )} δ 1 + ( + 1)δ

10 654 K.Ameeal Bibi ad M.Devi BM 0 ( S1, ) = μ + A (v) + μ + B (v, u 3 ) + μ + A (u 3 ) ={ μ A + (u i ) ( + 1)δ 1 /1 i } + Max{μ + A (v), μ + A (u 3 )} Mi{μ + A (v), μ + A (u 3 )} δ 1 + ( + 1)δ To fid k : Case (iv) for 4 i Bm ( S ) = μ A (v) + μ B (v, u i ) + μ A (u i ) 0 1, ={ μ A (u i ) ( + 1)δ 1 /1 i } + Max{μ A (v), μ A (u i )} Mi{μ A (v), μ A (u i )} δ δ BM ( S ) = μ + A (v) + μ + B (v, u i ) + μ + A (u i ) 0 1, ={ μ A + (u i ) ( + 1)δ 1 /1 i } + Max{μ + A (v), μ + A (u i )} Mi{μ + A (v), μ + A (u i )} + δ δ Hece the Star graph S1, is a Iterval valued fuzzy Bi-Magic labeled graph for 4. Propositio 3.8 Cosider a Path graph ( 4) with the fuzzy labelig of vertices ad edges which is trasformed ito a Iterval valued fuzzy Bi-Magic graph. Proof: I Path graph ( 4), for vertex v 1 ad edge v 1 v, we allocate δ = 0.0 ad for the rest of the vertices ad edges we allocate δ = Addig these values of δ to cocer vertices ad edges, we get two costat values which will be assumed as a upper limit for the iterval. (ie)., μ A + (v i ) + μ B + (v i, v j ) + μ A + (v j ) =k1 ad k for all v 1, v, v 3, V ad o multiplyig all the upper limit by 0.1, we get, μ A (v i ) + μ B (v i, v j ) + μ A (v j ) =k1 ad k for all v 1, v, v 3, V which satisfy the coditios of Bi-Magic labelig of a Iterval valued fuzzy graph (We are choosig arbitrarily the legth of the iterval).

11 Bi-Magic labelig of Iterval valued Fuzzy Graph 655 Example 3.9 Cosider the Path graph P5 (here =5) with fuzzy labeled which is ot Bi-Magic. Assig δ = 0.0 for vertex v 1 ad edge v 1 v, ad assig δ = 0.01 for the rest of the vertices ad edges The we obtai, μ A + (v 1 ) + μ B + (v 1, v ) + μ A + (v ) = 1.75 μ A + (v i ) + μ B + (v i, v j ) + μ A + (v j ) = 1.73 for i=,3,4,5 ad j=3,4,5 Which is labeled Bi-Magic. Now, o multiplyig all the upper limits by 0.1, we get μ A (v 1 ) + μ B (v 1, v ) + μ A (v ) = μ A (v i ) + μ B (v i, v j ) + μ A (v j ) = for i=,3,4,5 ad j=3,4,5 Which is labeled Bi-Magic. Theorem 3.10 Ay Fuzzy labeled graph ca be coverted ito a Iterval valued fuzzy Bi-Magic graph but the Iterval membership will ever be mutually disjoit. Proof: We kow that fuzzy labeled graph assigs some membership value for its vertices ad edges which is bijective. Thus, we are addig some δ correspodig to all vertices ad edges, we get the Bi-Magic sum for each pair of vertices ad coected edges. We take up that sum as the upper limit for the itervals. Now, we just multiply all the foud upper limits by 0.1, we get the lower limit of the iterval (see membership values i Fig 3). Thus, we obtaied iterval valued fuzzy graph which satisfies the coditio of Bi-Magic labelig. But the resulted iterval eed ot to be disjoit because we are choosig the legth of the iterval arbitrarily.

12 656 K.Ameeal Bibi ad M.Devi 4. CONCLUSION I this Paper, the cocept of a Iterval valued fuzzy Bi-Magic labelig has bee itroduced. Iterval valued fuzzy Bi-Magic labelig for Cycle, Star ad Path graphs have bee discussed. We further exteded this study o some more special classes of graphs. REFERENCES [1] Akram.M ad Dudek W. A., Iterval-valued fuzzy graphs, Computers ad Mathematics with Applicatios 61 (011) [] Gorzalczay M. B., A method of iferece i approximate reasoig based o iterval- valued fuzzy sets, Fuzzy Sets Syst. 1 (1987) [3] Gorzalczay M. B., A iterval-valued fuzzy iferece method some basic properties, Fuzzy Sets Syst. 31 (1989) [4] Hogmei.J ad Liahua.W, Iterval-valued fuzzy sub-semi groups ad subgroups associated by iterval-valued fuzzy graphs, 009WRI Global Cogress o Itelliget Systems (009) [5] A. M. Ismayil ad A. M. Ali, O Complete iterval-valued ituitioistic fuzzy graph, Advaces i fuzzy sets ad systems 18 (1) (014) [6] Mishra S.N,Aita pal,magic labelig of Iterval valued fuzzy graphs, Aals of fuzzy Mathematics ad Iformatics,Vol.11,(Feb 016),73-8 [7] Mordeso J. N. ad C. S. Peg, Operatios o fuzzy graphs, Iformatio Sci. 79 (1994) [8] Nagoor Gai. A, M. Akram ad D. R. Subahashii, Novel properties of fuzzy labelig graphs, Hidawi Publishig Corporatio Joural of Mathematics 014 (014) Article ID [9] Nagoor Gai. A ad D. Rajalaxmi, A ote o fuzzy labelig, Iteratioal Joural of Fuzzy Mathematical Archive 4 () (014) [10] Pal.M.,, Samata.S., ad Rashmalou.H, Some Results o Iterval-Valued Fuzzy Graphs, Iteratioal Joural of Computer Sciece ad Electroics Egieerig 3 (3) (015) [11] Talebi A.A., Rashmalou.H, Isomorphism o iterval-valued fuzzy graphs, A, fuzzy math. Iform. 6 (1) (013)