THE TOTAL IRREGULAR VALUE FOR DOUBLE LAYERED NEIGHBOURLY IRREGULAR FUZZY CHEMICAL GRAPHS

In this paper, using Neighbourly Irregular Fuzzy Graph and Neighbourly Irregular Chemical Graph, we introduce Neighbourly Irregular Fuzzy Chemical Graphs(NIFCG). We also discuss about the Total Irregular value for Double Layered Neighbourly Irregular Fuzzy Chemical Graphs(NIFCG) using edge cut method.


1.INTRODUCTION
A. Rosenfeld [1975] [7] A.Kauffman [1973] along with Yeh and Bang introduced the concepts of fuzzy graph. Inspired the papers of A.Nagoorgani [1] ,who introduced notations and some basic definitions for fuzzy graphs and fuzzy irregular graphs. J.Arockia Aruldoss [2] who introduced Neighbourly Irregular Chemical graphs inspired by S.Gnana Bhragsam [5] who introduced the NI graphs.
Throughout the paper we considered Neighbourly Irregular fuzzy chemical graph(NIFCG) and we used edge cut method to find The Total Irregular Value For Double Layered Neighbourly Irregular Fuzzy Chemical Graphs. Section two contains basic definitions and we introduce a new definitions for Neighbourly Irregular and Neighbourly Total Irregular Double Layered Fuzzy Chemical Graphs, in section three presents the theoretical concepts and finally we give conclusion on Total Irregular Value For D L NIFCG.

BASIC DEFINITIONS AND NEIGHBOURLY IRREGULAR, NEIGHBOURLY TOTAL IRREGULAR DOUBLE LAYERED FUZZY CHEMICAL GRAPHS DEFINITONS
A not empty set S has a fuzzy subset of mapping :

Definition
A Fuzzy graph is a pair of functions G: (,) where V is a fuzzy subset of , a symmetrical fuzzy relation  is on  . ie, : V  [0,1] and :

Definition
Let G = (,) where (u)   (u)for all uV and  (u,v)   (u,v)  u,v  V which is said to be the fuzzy graph H = (,) is called fuzzy Subgraph.

Definition
Let G = (,) is a fuzzy graph, if there is some vertices with distinct degrees which is adjacent to the vertex is called Irregular Graph.

Definition
Let G = (,) be a fuzzy graph, if there is a vertex which is adjacent to vertices with distinct total degrees where G is Total Irregular.

Definition
A Neighbourly Total Irregular Fuzzy Graph is said to be G = (,) if every two adjacent vertices of fuzzy graph have different total degrees.

Definition
Let V be a non empty set of vertices. A Neighbourly Irregular Double Layered Fuzzy Graph is a two identical function D L (G) = ( DL , DL ) [3] where fuzzy subset of V is DL, DL is a symmetric Double Layered fuzzy relation of DL. ie, DL:

Definition
Let DL(G) = (DL,DL) be a Double Layered Fuzzy graph. The total degree of a vertex uV is defined by td DLG (u) = d DLG (u)+ DLG (u) [6]. If each vertex of D LF (G) has the distinct degrees. This is known as Neighbourly Total Irregular Double Layered Fuzzy Graphs.

Example
Neighbourly Total Irregular Double Layered Fuzzy Graph

Definition
An edge cut is a set of edges of DLFG which is removing on edges to obtain the required graph. It is denoted by ie, D L (G)-e j, for some j's such that not all e j 's are removed.

Definition
Let V be a non empty set of atoms of corresponding molecules of Neighbourly Irregular Chemical

Definition
Let V be a non empty set of atoms of corresponding molecules of Neighbourly Irregular Double Layered Fuzzy Chemical Graph [8].

Definition
Let DL(G) = (DL,DL) be a Double Layered fuzzy graph. The total valencies of an atom of corresponding molecules of total NI(DL )FCG u belongs to V is stated as tdDLFC (u) = dDLG(u)+  DLFC(u). If every two atoms of DLFC(G) has the distinct total valencies [6]

Example
Sicl 4 ( silicon tetrachloride) [2] Neighbourly Total Irregular Double Layered Fuzzy Chemical Graph 2.12 Definition An edge cut is a set of edges of a Double Layered Chemical Graph which is by removing on edges to obtain the required graph. It is denoted by D LCF (G)-e j for some j's such that not all ej's are removed.

Example
Tecl4 (Tellurium tetrachloride) [4] Neighbourly Irregular Fuzzy Chemical Graph For any two adjacent atoms with different valencies, its total valencies are also different, then  DLFC is a constant function .The above result is satiesfied for every pair of adjacent atoms in D LFC (G).
Therefore every Neighbourly Irregular Double Layered Fuzzy Chemical Graph have a Neighbourly Total Irregular Double Layered Fuzzy Chemical Graphs for using edge cut method.

4.Conclusion
In this paper proposes the Total Irregular value for Double Layered Neighbourly Irregular Fuzzy Chemical Graphs using edge cut method. And we define some new definitions on Double Layer Fuzzy Chemical Graphs. Further work can be done by using Vertex cut method in Double Layered Fuzzy graph in NIC Graphs.