Properties of Fuzzy Labeling Graph

Transcription

1 Applied Mathematical Sciences, Vol. 6, 2012, no. 70, Properties of Fuzzy Labeling Graph A. Nagoor Gani P.G& Research Department of Mathematics, Jamal Mohamed College (Autono), Tiruchirappalli , Tamilnadu, India. D. Rajalaxmi (a) Subahashini Department of Mathematics, Saranathan college of Engineering, Tiruchirappalli , Tamilnadu, India Abstract In this paper a new concept of fuzzy labeling is introduced. A graph is said to be a fuzzy labeling graph if it has fuzzy labeling. Fuzzy sub graph, union, fuzzy bridges, fuzzy end nodes, fuzzy cut nodes and weakest arc of fuzzy labeling graphs have been discussed. And number of weakest arc, fuzzy bridge, cut node and end node of a fuzzy labeling cycle has been found. It is proved that (G ω ) is a fuzzy cut node and δ(g ω ) is a fuzzy end node of fuzzy labeling graph. Also it is proved that If G ω is a connected fuzzy labeling graph then there exists a strong path between any pair of nodes. Mathematics Subject Classification: 03E72, 05C72, 05C78 Keywords: fuzzy bridge, fuzzy cut node, fuzzy labeling graph 1. Introduction The first definition of fuzzy graph was introduced by Kaufmann (1973), based on Zadeh s [11] fuzzy relations (1971). A more elaborate definition is due to Azriel Rosenfeld [8] who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graph in During the same time Yeh and Bang [10] have also introduced various connectedness concepts in fuzzy graph. Till now fuzzy graphs has been witnessing a tremendous growth and finds applications in many branches

2 3462 A. Nagoor Gani and D. Rajalaxmi (a) Subahashini of engineering and technology [5]. Rosenfeld has obtained several concepts like bridges, paths, cycles, and trees and established some of their properties. Fuzzy end nodes and cut nodes were studied by K.R.Bhutani et al [2, 3]. Bhattacharya has established some connectivity concepts regarding fuzzy cut nodes and fuzzy bridges titled Some remarks on fuzzy graphs [1]. A fuzzy graph is the generalization of the crisp graph. Therefore it is natural that many properties are similar to crisp graph and also it deviates at many places. In crisp graph, A bijection f: V E N that assigns to each vertex and/or edge if G= (V, E), a unique natural number is called a labeling. In this paper the basic definitions and symbols were followed as in [6, 7]. 2. Preliminaries Let U and V be two sets. Then ρ is said to be a fuzzy relation from U into V if ρ is a fuzzy set of U V. A fuzzy graph G = (σ, µ) is a pair of functions σ: V [0, 1] and µ: V V [0, 1], where for all u, v V, we have µ (u, v) σ (u) Λ σ (v). A path P in a fuzzy graph is a sequence of distinct nodes v 1, v 2,..., v n such that µ (v i, v i+1 ) > 0; 1 i n; here n 1 is called the length of the path P. The consecutive pairs (v i, v i+1 ) are called the edge of the path. A path P is called a cycle if v 1 =v n and n 3 and a cycle is called a fuzzy cycle if it contains more than one weakest arc. The strength of a path P is n defined as Λ µ (v i, v i+1 ). The strength of connectedness between u and v is i=1 defined as the maximum of the strength of all paths between u and v and is denoted by µ (u,v). An u-v path P is called a strongest u-v path if its strength equals µ (u,v). A fuzzy graph is connected if for every u, v in σ *, µ (u,v)> 0.An arc of a fuzzy graph is called strong if its weight is at least as great as the connectedness of its end nodes when it is deleted and an u-v path P is called strong path if path P contains only strong arcs[4]. An edge is called a fuzzy bridge of G if its removal reduces the strength of connectedness between some pair of nodes in G. Similarly a node is a fuzzy cut node of G = (σ, µ) if removal of it reduces the strength of connectedness between some other pair of nodes. A node v is called a fuzzy end node if it has exactly one strong neighbor in G. The degree of a vertex v is defined as d(v) = µ (v, u). u v u V 3. Fuzzy labeling for Subgraph and union 3.1 Definition: A bijection ω is a function from the set of all nodes and edges of G* to [0, 1] which assign each nodes σ ω (u), σ ω (v) and edge µ ω (u, v) a membership

3 Properties of fuzzy labeling graph 3463 value such that µ ω (u, v) < σ ω (u) Λ σ ω (v) for all u, v V is called fuzzy labeling. A graph is said to be a fuzzy labeling graph if it has a fuzzy labeling and it is denoted by G ω. 3.2 Example: Fig 1 illustrate a fuzzy labeling graph G ω = (σ ω, µ ω ) where σ ω = {v , v , v , v } and µ ω = {(v 1, v 2 ) 0.01, (v 2, v 3 ) 0.02, (v 3, v 4 ) 0.08,(v 4, v 1 ) 0.06}. Fig.1. A fuzzy labeling graph G ω = (σ ω, µ ω ) 3.3 Definition: A cycle graph G * is said to be a fuzzy labeling cycle graph if it has fuzzy labeling. 3.4 Definition: The fuzzy labeling graph H ω = (τ ω, ρ ω ) is called a fuzzy labeling sub graph of G ω = (σ ω, µ ω ) if τ ω (u) σ ω (u) for all u V and,,, for all u, v V. 3.5 Proposition: If (τ ω, ρ ω ) is fuzzy labeling sub graph of (σ ω, µ ω ) then,, for all u, v V Let G ω = (σ ω, µ ω ) be any fuzzy labeling graph and H ω = (τ ω, ρ ω ) be its subgraph. Let (u, v) be any path in G ω then its strength be,. Since H is a sub graph, τ ω (u) σ ω (u) and,,, which implies,, for all u, v V. 3.6 Proposition: Union of any two fuzzy labeling graphs G ω 1 and G ω 2 is also a fuzzy labeling graph, if the membership value of the edges between G ω 1 and G ω 2 are distinct. Let G ω 1 and G ω 2 be any two fuzzy labeling graphs with distinct arc weight between G ω 1 and G ω 2. Then,, be the fuzzy sets of V 1, V 2,X 1 and X 2 respectively. Therefore σ ω (x) = (x) if x V 1 - V 2 and µ ω (x,y) = (x, y) if x, y X 1 - X 2. Similarly for (x) and (x, y). And σ ω (x) = max {, } if x V 1 V 2 and µ ω (x,y) = max { (x, y), (x, y)} if x, y X 1 X 2. Hence clearly σ ω = ( ) and µ ω = ( ).

4 3464 A. Nagoor Gani and D. Rajalaxmi (a) Subahashini 4. Properties of fuzzy labeling cycle 4.1 Proposition: If G * is a cycle then the fuzzy labeling cycle G ω has exactly only one weakest arc. n Let G ω be a fuzzy labeling cycle and let µ ω (x, y) = Λ µ ω (x i, y i ). Since G ω has a i=1 fuzzy labeling, it will have only one arc with µ ω (x, y). If we remove µ ω (x, y) from G ω, it will not reduce the strength of connectedness, which implies µ ω (x, y) is a weakest arc. Hence there exists only one weakest arc in any fuzzy labeling cycle. 4.2 Corollary: No cycle is a fuzzy cycle in fuzzy labeling graph. 4.3 Proposition: Let G ω be a fuzzy labeling cycle such that G * is a cycle, then it has (n-1) bridges. Let G * be a cycle with fuzzy labeling. By proposition 4.1, we will get only one weakest arc. We know that weakest arc is not a fuzzy bridge. Which implies the removal of any arc except the weakest arc will reduces the strength of connectedness. Hence every fuzzy labeling cycle have (n-1) bridges. 4.4 Proposition [9]: Let G = (σ ω, µ ω ) be a fuzzy graph such that G * is a cycle. Then a node is a fuzzy cut node of G if and only if it is a common node of two fuzzy bridges. 4.5 Proposition: If G * is a cycle with fuzzy labeling then it has (n-2) cut nodes. By preposition 4.3, every fuzzy labeling cycle has (n-1) fuzzy bridges (ie) it will have only one weakest arc, say µ ω (u, v). Therefore other than σ ω (u) and σ ω (v), all the remaining (n-2) nodes are common node of two fuzzy bridge. Hence by proposition 4, fuzzy labeling cycle has (n-2) cut nodes. 4.6 Proposition: If G * is a cycle with fuzzy labeling then, the graph has exactly two end nodes. By Proposition 4.1, G ω has exactly only one weakest arc, say µ ω (u, v), which implies σ ω (u) and σ ω (v) are end nodes. Hence every fuzzy labeling cycle graph has exactly two end nodes. 4.7 Proposition: If G * is a cycle with fuzzy labeling then the node of G ω is either a cut node or end nodes. The proof is obvious by proposition 4.5 and Proposition: If G ω is a fuzzy labeling cycle graph, then every bridge is strong and vice versa. Let G ω be a fuzzy labeling cycle graph with n nodes. By proposition 4.1, G ω has exactly only one weakest arc and also by proposition 4.3, G ω has (n-1) fuzzy

5 Properties of fuzzy labeling graph 3465 bridges. Now we claim that all these (n-1) bridges are strong. Let us choose an edge (x i, x i+1 ) from (n-1) edges. Since G ω is a cycle there exist two paths between the nodes x i and x i+1. (ie) one path with µ(x i, x i+1 )> 0 and the other path with µ(x i, x i+n,, x i+1 )>0. Therefore µ ( x i, x i+1 ) = µ(x i, x i+1 ). Which implies (x i, x i+1 ) is a strong arc. By repeating this process for the remaining edges we will get (n-1) strong arcs and converse is obvious. 5. Fuzzy labeling with bridge and strong edge 5.1 Proposition[5]: The following statements are equivalent a) (x,y) is a fuzzy bridge b) (x, y) < µ (x,y) c) (x, y) is not a weakest arc of any cycle 5.2 Proposition: If G ω is a fuzzy labeling graph, then G ω has at least one fuzzy bridge. Let G ω be a fuzzy labeling graph. Choose an edge (x,y) such that µ ω (x, y) is the maximum in the set of all values of µ ω (x i, y i ) for all x i, y i V. Therefore µ ω (x, y) > 0 and there exist some edge (u, v) distinct from (x, y) such that µ ω (u, v) < µ ω (x, y). Now we claim that µ ω (x, y) is a fuzzy bridge. If we remove (x, y) from G ω, then in its sub graph H ω. We have (x, y) < µ ω (x,y). Hence by proposition 4.8, (x, y) is a fuzzy bridge. 5.3 Remark: The converse of the above preposition is not true. Proof is obvious. 5.4 Proposition: If G ω is a connected fuzzy labeling graph then there exists a strong path between any pair of nodes. Let G ω be a connected fuzzy labeling graph and let (u,v)be any pair of nodes. Which implies µ (u,v) > 0. Now choose any edge (u, w) in (u, v), if µ(u,w) = µ (u,w) then it is strong. Otherwise choose some other edge, say (u,x) which satisfies µ(u,x) = µ (u,x). By repeating this process we can find a path in (u, v) in which all arcs are strong. 5.5 Proposition: Every fuzzy labeling graph has atleast one weakest arc. Let G ω be a fuzzy labeling graph and let (u, v) be an edge of G ω such that µ ω (u, v) is the minimum of all other µ ω s. if we remove µ ω (u, v) from G ω it does not reduces the strength of any path. In other words, after the removal, in its subgraph H ω we ve µ ω (u,v) < (u, v). Which implies µ ω (u,v) is neither a fuzzy bridge nor a strong arc. Therefore it must be one of the weakest arcs. 5.6 Proposition: For any fuzzy labeling graph G ω, δ(g ω ) is a fuzzy end node of G ω such that the number of arcs incident on, δ(g ω ) is at least two. Let G ω be a fuzzy labeling graph and there exist at least one node v with degree

6 3466 A. Nagoor Gani and D. Rajalaxmi (a) Subahashini δ(g ω ). Which implies the arcs which are incident on v may have lower membership value and it is not possible to have all the arcs which are incident on v as the weakest arc. Therefore it must have a strong neighbor. Hence, δ(g ω ) is an end node of G ω. 5.7 Proposition: Every fuzzy labeling graph has at least one end nodes. It is trivial that for any fuzzy labeling graph there exists at least one node with degree δ(g ω ). Therefore by proposition 5.6, δ(g ω ) is an end node of G ω. 5.8 Proposition: Every fuzzy labeling graph has at least one cut node. For any fuzzy labeling graph G ω, we can find at least one node with maximum degree (G ω ). Let v be a node with degree (G ω ) which implies the edges which are incident in v may have higher membership value. Therefore removal of such a node v will reduces the strength of connectedness. Hence v is a cut node. References [1] P.Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letters 6 (1987) [2] K.R.Bhutani, J.Moderson and A. Rosenfeld, on degrees of end nodes and cut nodes in fuzzy graphs, Iranian Journal of fuzzy systems, vol. 1, No. 1 (2004) pp [3] K.R.Bhutani and A. Rosenfeld, fuzzy end nodes in fuzzy graph, information sciences, 152: , June [4] K.R.Bhutani and A. Rosenfeld, strong arcs in fuzzy graph, Elactronic notes in discrete mathematics, 15(2003), [5] G.J. Klir, Bo Yuan, fuzzy sets and fuzzy logic, Theory and applications. PHI (1997) [6] John N.Mordeson and Premchand S.Nair, Fuzzy Graphs and Fuzzy hypergraphs, Physica Verlag, Heidelberg [7] A.Nagoorgani, V.T. Chandrasekaran, A first look at fuzzy graph theory, Allied Publishers Pvt. Ltd, [8] A. Rosenfeld, Fuzzy Graph, In: L.A. Zadeh, K.S. Fu and M.Shimura, Editors, Fuzzy sets and their Applications to cognitive and decision Process, Academic press, New York (1975) [9] M.S.Sunitha, A.Vijayakumar, A characterization of fuzzy trees, Information sciences 113(1999) [10] R.T.Yeh and S.Y.Bang, Fuzzy relations, fuzzy graphs and their applications to Clustering analysis. In: L.A.Zadeh, K.S.Fu and M.Shirmura, Editors, Fuzzy sets and Their Applications, Academic press (1975), pp [11] L.A. Zadeh, Fuzzy sets, Information and control 8(1965), Received: February, 2012