FUZZY GRAPHS ON COMPOSITION, TENSOR AND NORMAL PRODUCTS

Transcription

1 International Journal of Scientific and Research Publications, Volume 2, Issue 6, June Abstract FUZZY GRAPHS ON COMPOSITION, TENSOR AND NORMAL PRODUCTS Dr. G.NIRMALA* and M.VIJAYA** *Associate Professor, P.G & Research Department of Mathematics, K.N.G Arts College for Women (Autonomous), Thanjavur Tamil Nadu, India nirmalamanohar11@yahoo.com **Head, Department of Mathematics, Marudupandiyar College, Thanjavur , Tamil Nadu, India. mathvijaya79@yahoo.com Special fuzzy graph can be obtained from two given fuzzy graphs using the operations, Cartesian product, composition, tensor and normal products. In this paper, we find the degree of a vertex in fuzzy graphs formed by these operations in terms of the degree of vertices in the given fuzzy graphs in some particular cases. Index Terms- Cartesian product, Composition, Degree of a vertex, Tensor product, Normal product. 1. Introduction Fuzzy graphs introduced by Rosenfeld in 1975[1-2,9,10]. The operations of union, join, Cartesian product and composition on two fuzzy graphs were defined by Moderson.J.N. and Peng.C.S [3-8]. In this paper, we study about the degree of a vertex in fuzzy graphs which are obtained from two given fuzzy graphs using the operations Cartesian product and composition of two fuzzy graphs, tensor and normal product of two fuzzy.in general, the degree of vertices in Cartesian product and composition of two fuzzy graphs, tensor and normal product of two fuzzy graphs G 1 andg 2 cannot be expressed in terms of those in G 1 andg 2. In this paper, we find the degree of vertices in Cartesian product, composition, tensor and normal product of G 1 andg 2 in some particular cases. 2. Research Elaborations Definition 2.1: A fuzzy subset of a set V is a mapping σ from V to [0, 1]. A fuzzy graph G is a pair of functions G: (σ, μ) where σ is a fuzzy subset of a non empty set V and μ is a symmetric fuzzy relation on σ, (i.e.) μ (uv) σ (u) σ (v). The underlying crisp graph of G: (σ, µ) is denoted by G*: (V, E) where E V V. Definition 2.2: Let G: (σ, µ) be a fuzzy graph. The degree of a vertex u in G is defined by d G (u) = ( ) = ( ) Definition 2.3: The order of a fuzzy graph G is defined by O (G) = ( ) Note: Throughout this paper G₁: (σ₁, μ₁) and G₂ :( σ₂, μ₂) denote two fuzzy graphs with underlying crisp graphs G₁*: (V₁, E₁) and G₂* :( V₂, E₂) with V i =P i, i = 1, 2. Also d (G i *) (u i ) denotes the degree of u i in G i * Definition 2: The cartesian product of two fuzzy graphs G₁ and G₂ is defined as a fuzzy graph G = G₁ x G₂ : ( σ₁x σ₂, μ₁ x μ₂ ) on G*: (V, E) where V=V₁ x V₂ and E= {((u₁,u₂)(v₁,v₂)) / u₁=v₁, u₂v₂ E₂ or u₂=v₂,u₁v₁ E₁} with (σ 1 x σ 2 )(u 1, u2) = σ₁(u₁) σ 2 (u 2 ), for all (u 1,u 2 ) V₁ V₂ ₁( ₁) ₂ ( ₂ ₂) ₁ ₁ ₂ ₂ ₂ & (µ 1 µ 2 )(( u 1,u 2 )( v 1,v 2 )) = { ₂( ₂) ₁)( ₁ ₁) ₂ ₂ ₁ ₁ ₁ Definition 2: The Composition of two fuzzy graphs G₁ and G₂ is defined as a fuzzy graph G= G 1 [ G₂ ] : ( σ₁ σ₂, μ₁ μ₂ ) on G*:(V,E) where V= V₁ V₂ and E={((u 1,u 2 )(v 1,v 2 )) / u 1 =v 1, u 2 v 2 E 2 or u 2 v 2,u₁v₁ E 1 } with (σ 1 x σ 2 )(u 1, u2) = σ₁(u₁) σ 2 (u 2 ), for all (u 1,u 2 ) V₁ V₂

2 International Journal of Scientific and Research Publications, Volume 2, Issue 6, June & (µ 1 µ 2 )(( u 1,u 2 )( v 1,v 2 )) ={ ₁( ₁) ₂ ( ₂ ₂) ₁ ₁ ₂ ₂ ₂ ₂( ₂) ₁ ( ₁ ₁) ₂ ₂ ₁ ₁ ₁ ₂( ₂) ₂( ₂) ₁ ( ₁ ₁) ₂ ₂ ₁ ₁ ₁ 3. Results Definition 3.1: The normal product of two fuzzy graphs (σ i, µ i ) on G i = (V i, X i ), i=1,2 is defined as a fuzzy graph (σ 1 σ 2, µ 1 µ 2 ) on G = (V,X) where V= V 1 X V 2 and X ={((u,u 2 )(u,v 2 ) \ u V 1, (u 2 v 2 ) X 2 } {((u 1,w)(v 1, w))\ (u 1,v 1 ) X 1,w V 2 } {(u 1, u 2 )(v 1,v 2 )\(u₁v₁) X₁,(u₂v₂) X₂} Fuzzy sets σ 1 σ 2 and µ 1 µ 2 are defined as (σ 1 σ 2 ) (u 1, u 2 ) = (σ 1 (u 1 ) σ 2 (u 2 ) ) (u 1, u 2 ) V 1 xv 2 ( 1 2) ((u, u 2 )(v 1, v 2 )) = { 1 (u) 2 (u 2 v 2 ) } u V 1 and (u 2 v 2 ) X 2 ( 1 2 ) ((u 1, w)(v 1, w)) = { 1 (u 1 v 1 ) 2(w)} w V 2 and (u 1 v 1 ) X 1 (µ1 2) ((u 1, u₂)(v₁, v 2 )) = {µ 1 (u₁v₁) µ 2 (u₂v 2 )} (u 1 u 2 ) X 1 and (v 1 v 2 ) X 2 Definition 3.2: The tensor Product of two fuzzy graphs (σ i, µ i ) on G i = (V i, X i ), i=1,2 is defined as a fuzzy graph (σ₁ σ 2, µ 1 µ 2 ) on G = (V,X) where V= V 1 X V 2 and X ={(u 1, u 2 ), (v 1, v 2 )\(u₁,v₁) X₁,(u₂,v₂) X₂}. Fuzzy sets σ 1 σ 2 and µ₁ µ 2 are defined as ( 1 2) (u 1, u 2 ) = {( 1 (u 1 ) 2 (u 2 )} for all (u 1, u 2 ) V ( 1 2) {(u 1, u 2 ), (v 1, v 2 )} = * 1 (u 1, v 1 ) 2 (u 2, v 2 )} (u 1, u 2 ) X 1 and (v 1, v 2 ) X Degree of a vertex in Cartesian product By the definition, for any vertex (u 1, u 2 ) V 1 x V 2, d G ₁ xg₂ (u₁ u₂) = (µ₁ µ₂)(( ₁ ₂)( ₁ ₂)) ( )( ) = σ₁( ₁) µ₂( ₂ ₂) + σ₂( ₂) µ₁( ₁ ₁) In the following theorems, we find the degree of (u 1, u2) in G 1 X G 2 in terms of those in G 1 and G 2 in some particular cases. Theorem 3.1: Let G 1 :(σ₁,µ₁) and G 2 :(σ₂,µ₂) be two fuzzy graphs. If σ 1 µ 2 and σ 2 µ 1 then d G₁ x G₂ (u₁,u₂) = d G₁(u₁) +d G₂(u₂). Proof: From the definition of a degree of a vertex in cartesian product d G₁ x G₂ (u₁,u₂) = ₁ ₁ ₂ ₂ ₂ σ₁( ₁) µ₂( ₂ ₂) + ₂ ₂ ₁ ₁ ₁ σ₂( ₂) µ₁( ₁ ₁) = ₂ ₂ ₂ µ₂( ₂ ₂) + ₁ ₁ ₁ µ₁( ₁ ₁) (since σ 1 µ 2 and σ 2 µ 1 ) = d G₁ (u 1 ) +d G₂ (u 2 ).

3 International Journal of Scientific and Research Publications, Volume 2, Issue 6, June Example 3.1 u₁ (.6) u₁ (.7) u₂ () v₂ (.7) G 2 σ₁ µ₂ σ₂ µ₁ G 1 (u₁, u₂) (u₁, v₂).6 (v₁, u₂) (v₁, u₂).7 Figure. 1. Cartesian product of G₁ & G₂ (G₁ x G₂) Here G₁, G₂ are two fuzzy graphs d G₁ x G₂ (u₁,u₂) = d G₁(u₁) + d G₂ (u 2 ). ( since σ₁ µ₂, σ₂ µ₁ & by theorem 3.1 ) = + =.9 d G₁ x G₂ (u₁,v₂) = + =.9 Similarly we can find the degrees of all the vertices in G₁ x G₂ This can be verified in the figure Degree of a vertex in composition By the definition, for any vertex (u 1,u 2 ) V 1 x V 2, d G₁ x G₂ (u 1, u 2 ) = ( ₁ ₂)( ₁ ₂) ( µ₁ µ₂)(( ( ₁ ₂)( ₁ ₂)) = ₁ ₁ ₂ ₂ ₂ σ₁( ₁) µ₂( ₂ ₂) + ₂ ₂ ₁ ₁ ₁ σ₂( ₂) µ₁( ₁ ₁) + ₂ ₂ ₁ ₁ ₁ σ₂( ₂) µ₁ (u₁ v₁) Theorem 3.2 Let G 1 : (σ₁, µ₁) and G 2 :(σ 2,µ 2 ) be two fuzzy graphs. If σ 1 µ 2 and σ 2 µ 1 then d G₁[G2] (u₁,u₂) =P₂ d G₁(u₁) +d G₂(u₂). Proof: d G₁ [G₂] (u 1, u 2 ) = ₁ ₁ ₂ ₂ ₂ σ₁( ₁) µ₂( ₂ ₂) + ₂ ₂ ₁ ₁ ₁ σ₂( ₂) µ₁( ₁ ₁) + ₂ ₂ ₁ ₁ ₁ σ₂( ₂) µ₁ (u₁v₁) = ₂ ₂ ₂ µ₂( ₂ ₂) + ₂ ₂ ₁ ₁ ₁ µ₁( ₁ ₁) + ₂ ₂ ₁ ₁ ₁ µ₁ ( ₁ ₁) (since σ₁ µ₂ and σ₂ µ₁) = d G₂ (u 2 ) + V 2 ₁ ₁ ₂ µ₁( ₁ ₁) = d G₂ (u 2 ) + p₂ d G₁(u₁). Example 3.2

4 International Journal of Scientific and Research Publications, Volume 2, Issue 6, June u₁ (.6) Consider the fuzzy graphs G₁, G₂ and G₁ [G₂] in figure 2. u₁ (.7) G 1 u₂ () v₂ (.7) G 2 (u₁, u₂) (u₁, v₂).7 (v₁, u₂) (v₁, u₂).7 Figure.2. Composition of G₁ & G₂ (G₁ [G₂]) Here σ₁ µ₂ and σ₂ µ₁ and by theorem 3.2 d G₁ [G₂] (u₁,u₂) = d G₂(u₂) +P₂d G₁(u₁). = + 2 () =1 d G₁ [G₂] (u₁,v₂) = d G₂(v₂) +P₂d G₁(u₁). = + 2() = 1 Similarly we can find the degrees of all the vertices in G₁[ G₂] This can be verified in the figure Degree of a vertex in tensor product By definition, for any (u₁, u₂) V₁ V₂ d G₁ G₂ (u₁, u₂) = (µ₁ µ₂)(( u₁, u₂)( v₁, v₂)) = ₁ ₁ ₁ µ₁ ( ₁ ₂) µ₂ ( ₂ ₂) Theorem 3.3: d G₁ G₂ (u₁, u₂) Let G₁ : (σ₁, µ₁) and G₂ : (σ₂, µ₂) be two fuzzy graphs. If µ₂ µ₁ then = d G₁(u₁).And if µ₁ µ₂ then d G₁ G₂ (u₁, u₂) = d G₂(u₂). Proof: d G₁ G₂ (u₁, u₂) = µ₁ ( ₁ ₂) µ₂ ( ₂ ₂) ₁ ₁ ₁ = µ₁ (u₁, v₁) = d G₁(u₁)

5 International Journal of Scientific and Research Publications, Volume 2, Issue 6, June Example 3.3 u₁ (.9).6 u₂ (.9).7 ( u₁, u₂)( 9) ( u₁, v₂)( 8).6.6 v₁ (.7) v₂ (.8) ( ₁, u₂)( 7) (v₁, v₂)( 7) G₁ G₂ G₁ G₂ Figure. 3. Tensor product of G₁& G₂ (G₁ G₂) Consider the fuzzy graphs G₁ and G₂ in fig 3. Here µ₂ µ₁ and by theorem 3.3 d G₁ G₂ (u₁, u₂) = d G₁(u₁) =.6 d G₁ G₂(v₁,v₂) = d G₁(v₁) =.6 This can be verified in the fig 3. 3 Degree of a vertex in normal product By definition, for any (u₁, u₂) V₁ V₂ d G₁ G₂ (u₁, u₂) = (( ₁ ₁)( ₂ ₂)) ( ₁ ₂)(( ₁ ₁)( ₂ ₂) = ₁ ₁ ₂ ₂ ₂ σ₁( ₁) µ₂ ( ₂ ₂) σ₂( ₂) µ₁ ( ₁ ₁) ₂ ₂ ₂ ₁ ₁ ₁ µ₂( ₂ ₂) µ₁( ₁ ₁) Theorem 3 Let G 1 : (σ₁,µ₁) and G 2 :(σ 2,µ 2 ) be two fuzzy graphs. If σ 1 µ 2, σ 2 µ 1 and µ₁ µ₂ then d G₁ G₂(u 1,u 2 ) =P 2 d G1 (u 1 ) +d G2 (u 2 ). Proof: d G₁ G₂ (u₁, u₂) = ( ₁ ₁)( ₂ ₂) ( ₁ ₂)( ₁ ₁)( ₂ ₂) = ₁ ₁ ₂ ₂ ₂ σ₁( ₁) µ₂ ( ₂ ₂) σ₂( ₂) µ₁ ( ₁ ₁) ₂ ₂ ₂ ₁ ₁ ₁ µ₂( ₂ ₂) µ₁( ₁ ₁) = ₂ ₂ ₂ µ₂( ₂ ₂) + ₂ ₂ ₁ ₁ ₁ µ₁( ₁ ₁) + ₁ ₁ ₁ µ₁ ( ₁ ₁) (since σ₁ µ₂, σ₂ µ₁ and µ₁ µ₂) = d G₂ (u 2 ) + V 2 d G₁(u₁). = d G₂ (u 2 ) + p₂ d G₁(u₁).

6 International Journal of Scientific and Research Publications, Volume 2, Issue 6, June Example 3 u₁ (.6).3 v₁ (.7) G 1 u₂ () v₂ (.7) G 2 (u₁, u₂) (u₁, v₂) (v₁, u₂) G₁ G₂ (v₁, v₂).7 Figure. Normal product of G₁ & G₂ (G₁ G₂) Consider the fuzzy graphs G₁ and G₂ in figure 4. Here σ 1 µ 2, σ 2 µ 1 and µ₁ µ₂ and by theorem 3 d G₁ G₂(u₁,u₂) = d G₂(u₂) + P₂d G₁(u₁) = + 2(.3) =1 d G₁ G₂(v₁,v₂) = d G₂(v₂) + P₂d G₁(v₁) = + 2(.3 ) = 1 Similarly we can find the degrees of all the vertices in G₁ x G₂ This can be verified in the figure 3. Conclusion In this paper,we have found the degree of vertices in G₁ x G₂, G₁[G₂], G₁ G₂, G₁ G₂ in terms of the degree of vertices in G₁ and G₂ under some conditions and illustrated them through examples. This will be helpful when the graphs are very large. Also they will be very useful in studying various properties of cartesian product, composition, tensor product, normal product of two fuzzy graphs. REFERENCES [1] R.Balakrishnan and K.Ranganathan, A text book of Graph Theory, Spinger, [2] A.Roseneld, fuzzy graphs, in: L.A.Zadeh, K.S. Fu,, K.Tanaka, M.Shimura (Eds), Fuzzy sets and Their Applications to Cognitive and Decision Processes, Academic Press, New York,1975,pp [3] J.N.Mordeson, C.S.Peng, Operation on fuzzy graphs. Information Sciences 79 (1994) [4] J.N.Mordeson, & P.S.Nair, Information Sciences 90 (1996) [5] John N.Mordeson and Premchand S.Nair, Fuzzy Graphs and Fuzzy hypergraphs, Physica Verlag, Heidelberg [6] A.Nagoorgani and V.T.Chandrasekaran, Fuzzy graph Theory, Allied Publishers pvt. Ltd

7 International Journal of Scientific and Research Publications, Volume 2, Issue 6, June [7] A.Nagoorgani & M.Basheer Ahamed, 2003, Order and Size of fuzzy graphs, Bullet ion of pure and applied sciences. 22E (1), pp [8] A.Nagoorgani & K.Radha,2009, The degree of a vertex in some fuzzy graphs,international Journal of Algorithms,Computing and Mathematics,vol 2 ( ). [9] Harary. F,.Graph Theory, Addition Wesly, Third printing, October [10] Zimmermann, H.J., Fuzzy Set Theory and its applications,kluwer-nijhoff,boston,1985. AUTHORS Dr. G. NIRMALA Associate Professor, P.G & Research Department of Mathematics, K.N.G Arts College for Women (Autonomous), Thanjavur Tamil Nadu, India. nirmalamanohar11@yahoo.com Correspondence Author: M.VIJAYA Head, Department of Mathematics, Marudupandiyar College, Thanjavur , Tamil nadu, India. mathvijaya79@yahoo.com M. No: