NORDHAUS-GADDUM INEQUALITIES FOR ANTI FUZZY GRAPH

The objective of this paper is to finds the lower and upper bounds of Nordhaus-Gaddum inequalities of fuzzy chromatic number for anti-fuzzy graph. This paper analyzes the chromatic index of complementary anti fuzzy graphs in some cases. A theorem is proved for anti-fuzzy graph to be k-critical. Examples are provided to derive the vertex coloring of these graphs.


INTRODUCTION
A mathematical representation helps to make out the problem in a difficult situation. The best reachable solution is to convert the difficulties into graph. Fuzzy graph theory has used to representation of many decision making problems in vague environment which have several applications. The main part of the difficulty is considered as vertices and their connection between theses vertices are considered as edges. These are represented with fuzzy value [o to 1] to determine the vagueness. Some time, vagueness exists in a relation that attains maximum value. This kind of form is known as Anti Fuzzy Graph. Graph coloring is to assign colors to certain elements of a graph subject to constraints. Vertex coloring is the most common graph coloring technique which used in various research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking etc., In 1956, [7] [11]  To determine chromatic number of AFG, family of -cuts of is determined and then the chromatic number of each is determined by using crisp k-vertex coloring k . The Chromatic number of is defined through a monotone family of sets. [11]

CHROMATIC NUMBER OF SELF COMPLEMENTARY ANTI FUZZY GRAPH
This section observes the features of selfcomplementary anti fuzzy graph and derives the fuzzy chromatic number for complement of antifuzzy graph.

Proposition 4.1
For a self-complement AFG, the underlying graph and its complement must be strong. But every strong AFG need not be self-complementary.

Proposition 4.2
If self-complementary exists in AFG then it must be a v-nodal anti fuzzy graph with effective edges. Proof: Every n-vertex self-complementary graph has exactly n(n-1)/4 edges and it must have the diameter either 2 or 3. [5]In complementary AFG, the vertices are isomorphic to each other by the property of complement. That is, = .
To claim the isomorphism, (x,y) must be equal to 0. From this, (x,y) = Every (3x3 rook's graph) Paley graph is the best example of self-complementary graphs. Example 4.5 with ) = 0.8 for all provides the self-complement.

Proposition 4.3
A Self complementary graph doesn't exist in a complete anti fuzzy graph. Since the complement of complete anti fuzzy graph is a null graph. There is no possible to finds an isomorphism between them.

Proposition 4.4
If AFG preserves the self-complement, then their chromatic number are same. Converse part of this statement is not necessarily true. Proof: If AFG preserves self-complement then, GA( ( ). That is, there exists an isomorphism between obviously, (GA) = ( ).

Example-4.5 Computation of fuzzy chromatic index using the -cuts
That is their chromatic index,

CHROMATIC NUMBER OF COMPLETE ANTI FUZZY GRAPH AND ITS COMPLEMENT
This section observes the chromatic number of a complete AFG and compared this result with its complementary graph. From this observation, a circumstance is provided for the computation of chromatic number.

CONCLUSION
The chromatic number of an anti-fuzzy graph is derived based on vertex coloring. This paper concludes that the chromatic number of self-complementary graph is same. The sum and product of chromatic number of complete antifuzzy graph is determined. For , lower and upper bounds for the Nordhaus -Gaddum inequality is, n+1 ≤ + ( ) ≤ 2n and n ≤ ( ) ≤ n 2 .