FUZZY ANTI-MAGIC LABELING ON SOME GRAPHS

In this Paper, we introduced the concept of fuzzy anti-magic labeling in graphs. We defined Fuzzy Anti-Magic Labeling (FAML) for Cycle, Star, Path and Antiprism graphs. A fuzzy graph G: (σ, μ) is known as fuzzy anti-magic graph if there exists two bijective functions σ: V → [0,1] and μ: V × V → [0,1] such that μ u, v < σ(u)  σ(v) with the property that the sum of the edge labels incident to each vertex, the sums will all be different. We investigated and verified that fuzzy Cycle graphs, fuzzy Star graphs, fuzzy Path graphs and fuzzy antiprism graphs admits fuzzy anti-magic labeling.Further some properties related to fuzzy bridge and fuzzy cut vertex have been discussed.


INTRODUCTION
We begin with a finite, connected and undirected graph : ( , )without loops and multiple edges. Throughout this paper (G) and (G) denote the vertices and edges respectively. In recent years, graph theory has been actively implemented in the fields of Bio-chemistry, Electrical engineering,  (v) has been suggested by Zadeh (1). The theory of fuzzy graphs was independently developed by Rosenfeld, Yeh and Bang. Fuzzy graph theory is finding extensive applications in modeling real time systems where the level of information congenital in the system varies with different levels of precision.
An edge uv is called a fuzzy bridge of G, if its removal reduces the strength of connectedness between some pair of vertices in G. Equivalently (u,v) is a fuzzy bridge iff there are nodes x,y such that (u,v) is a arc of every strongest x-y path.A G :  ,   Nagoorgani et al. (2) introduced the concept of fuzzy vertex is a fuzzy cutvertex of if removal magic labeling and properties of fuzzy labeling. Akram et al. introduced interval valued fuzzy graphs, Strong intuitionistic fuzzy graphs, m-polar fuzzy graphs and novel properties of fuzzy labeling graphs (3).
Already we published two articles in fuzzy Bi-magic labeling and Interval valued fuzzy Bi-magic labeling (4,5). In this paper, we introduced the concept of Fuzzy anti-magic labeling on some standard graphs.

PRELIMINARIES AND OBSERVATIONS
Let U and V be two non-empty sets. Then is said to be a fuzzy relation from U into V if is a fuzzy set of UxV. A fuzzy graph : ( , ) is a pair of functions : → [0,1] and : × → [0,1]where for all u,v ∈ , we have , of it reduces the strength of connectedness between some pair of vertices in G. A fuzzy graph admits antimagic labeling,if the sum of the edge labels incident to each vertex, the sums all will be different and it is denoted by 0 . A fuzzy graph which admits an anti-magic labeling is called Fuzzy anti-magic labeling graphs.

Definition 3.1.
A Cycle or Circulant graph is a graph that consists of a single cycle. The number of vertices in a cycle graph Cn equals the number of edges and every vertex has degree 2.
A Cycle graph which admits fuzzy labeling is called a fuzzy labeling Cycle graph and anti-magic Here, we investigated the results for fuzzy ~ anti-magic cycle Am 0 (C 7 ) for n=7.

Case (i): i is even
Then i=2x for any positive integer x.
For each edge vi, vi+1, the fuzzy anti-magic labelings are as follows: If n is odd, then the cycle Cn admits a fuzzy anti-magic labeling.

Proof:
Let G be a cycle with odd number of vertices The fuzzy labeling is defined as follows: Am 0 (C 7 ) 6 6 7 7 Am 0 (C 7 )

Case (i): i is odd
Then i=2x-1 for any positive integer x.
For each edge vi, vi+1 , the fuzzy anti-magic labelings are as follows: Subcase (i): Here, we investigated the results for fuzzy anti-magic ~ 2 Therefore, from the above cases, we verified that Cn is a fuzzy anti-magic graph if Cn has odd number of vertices.

Definition 3.4.
A fuzzy Star graph consists of two vertex sets V and U with = 1 and > 1 such that   , > 0 and , In a fuzzy Star graph, if an anti-magic labeling exists then it is called a fuzzy anti-magic  1

~ labeling Star graph and it is denoted by
Am 0 (S 1,n ) . For n ≥ 3, the Star graph S1,n admits a fuzzy anti-magic labeling.

Proof:
Let S1,n be the Star graph with v,u1, u2,…un as vertices and vu , vu , ….vu as edges. Let → [0,1] such that 1 2 n   one can choose z=0.01 if n ≥ 3. The fuzzy labeling is defined as follows: A Path with atleast two vertices is connected and has two terminal vertices (vertices that have degree 1) while all others (if any) have degree 2.
In a graph which admits fuzzy labeling is called a fuzzy path graph and anti-magic labeling exists then it is called as a fuzzy anti-magic Path graph.
i=2x for any positive integer    Theorem: 3.9 For n ≥ 3, the Path graph Pn has (n-1) fuzzy anti-magic labeling.
Let P be a Path with length n ≥ 1 and v1,v2,v3,…vn and v1v2, v2v3,…,vn-1vn are the vertices and edges of P.
Let → [0,1] such that one can choose z=0.01 if n ≥ 3. If the length of the path P is Odd, then the fuzzy labeling is defined as follows: Here, we investigated the results for fuzzy anti-magic labeling of Path graph for n=5.

Case (i): i is even
Then i=2x for any positive integer x For each edge vi,vi+1 Subcase (i):
In view of the above labeling pattern, the edges are distinctly labeled in such a way that when taking the sum of the edge labels incident to each vertex, the sums will be different.

PROPERTIES OF FUZZY ANTI-MAGIC GRAPHS Proposition 4.1.
For every fuzzy anti-magic graph G, there Theorem 3.12.
exists atleast one fuzzy bridge.