 Open Access
 Authors : Nandhini. R , Akshaya. G , Kayalvizhi. M
 Paper ID : IJERTV9IS020053
 Volume & Issue : Volume 09, Issue 02 (February 2020)
 Published (First Online): 17022020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Total ZeroDivisor Graph of A Field
Nandhini. R, Akshaya. G, Kayalvizhi. M Thassim Beevi Abdul Kader College for Women, Kilakarai
Abstract: Let F be a Field with z(F), its set of zero divisors. The total zero divisor graph of F, denoted Z((F)) is the undirected (simple) graph with vertices Z()=Z(F){0}, the set of non zero, zero divisors of F, and for distinct x, y () the vertices x and y are adjacent if and only if x+ y z(F). In this paper, we study if Z((F)) is finite and every vertex of Z((F)) has a finite degree then F is finite and also prove that Z((F)) connected with diam3.

INTRODUCTION
In this paper, we study the total zero divisor graph is the (undirected) graph with vertices Z(F)=Z(F) {0}. The set of nonzero zero divisor of F and for distinct x, y z(F), the vertices x and y are adjacent if and only if x+ y z(F). It is denoted by Z((F)) and is the (induced) subgraph of total graph. We show that Z((F)) is finite then F is finite and not an integral domain, if every vertex of Z((F)) has finite degree then F is finite and also prove that Z((F)) is connected with diam3. For some other recent papers on zero divisor graphs.

Definition:
A ring R is called a coloring if (0()) is finite.

Definition:
An element xR is said to be a zero divisor if there exists some element 0yR such that xy=0.

Presumption:
(0()) = 1 if and only if R = {0}.

presumption:
(0()) = 2 if and only if R is an integral domain, R 4
or R 2[]()2.

Definition:
The chromatic number of a zerodivisor graph of a ring R is equal to the clique number of the ring. That is, (0())
=cl(R).
III. MAIN RESULT
2.1Definition:


PRELIMINARIES
3.1Theorem:
Let F be field then the total zero divisor graph if finite
The number of edges incident with a vertex Vis called the degree of V and it is denoted by d(V). The minimum and maximum degree of a vertex of a graph are respectively denoted by and .

Definition:
A graph G in which every vertex is adjacent to every other vertex is called a complete graph. Complete graph is represented as where n is the number of vertices in

Definition:
The chromatic number of a zerodivisor graph of a ring R, denoted by (0()) is the minimal number of colors required to assign each vertex in a zerodivisor graph a color so that no two adjacent vertices are assigned the same color.

Definiton:
A graph 0() is a kcolorable if 0() can be colored with less than or equal to k colors.

Defnition:
A graph G is said to be a connected graph. If there is at least one path between every pair of vertices in G. otherwise G is said to be a disconnected graph.

Definition:
if and only if either or an integral domain. In particular if 1 Z((F)) .Then F ia finite.
Proof;
Let F be a field and Z(F) be the set of zero divisors in F and Let Z((F)) be the total zero divisor graph. Then all vertices of Z((F)) is nonzero, zero divisor of F.
It is trivial that if F is finite then Z((F)) is also finite.
Suppose that Z((F) is finite and nonempty. This implies that Z(F) is finite, suppose these are two elements u, vF, u0,v0.suchthat u+vZ(F)
Let I=Ann(Z), then u+vI
Since u+v ()this implies that I () further I is finite and f(u+v) I for all fF.[u+v ,fFf(u+v) I]suppose F is finite.Then there is an iI suchthat K={fF/f(u+v)=i}is infinite.
For any f, tK
F(u+v) = i, t(u+v) = i (ft) (u+v) =0
(ft) Ann(u+v) {since, KAnn(u+v), K is ftJftVAnn(u+v) infinite)
Where kAnn(u+v) Since ftZ(F)
i.e. Ann(u+v) Z(F), is infinite, a contradiction therefore F must be finite.

Theorem:
Let f be a field with identity. Then S = F Ã— 0 is a field
Any two distinct vertices a and b in graph G, the distance between a and b, denoted by d (a, b) is the length of a shortest path connecting a and b.
without identity, S = Z(S), and
(S)
2
(F).
Proof: Clearly S = Z(S) and T has no identity. Define : F/
S / by ([u]) = [(u, 0)]. It is easily verified that (u)
= (v) for u, v F if and only if ((u, 0)) = ((v, 0)), and [(u, 0)] = [(u, 1)] for every u F. Thus, is a welldefined bijection. Moreover, restricts to a graph isomorphism from (F) to (S) since [(u, 0)] [(v, 0)] = [(0, 0)] if and only if [u][v] = [0].

Theorem:
Let F be a field such that Z(F) Is not an ideal of F then Z(F) is connected with diamZ(F)=2
Proof: Each u () is adjacent to 0. Thus, u0v is a path in Z(F) of length two between any two distinct u, v () Moreover, there are nonadjacent u, v () since Z(F) is not an ideal of F.
So, diamZ(F)=2.
Hence proved

Theorem: Let F be a field then Z(F) is connected with diam3
Proof:
Let u, v be vertices in Z((F)),
There exists u+z Z(F), v+WZ(F)
If u+v Z(F) then uv is a path of length is perpendicular containing u, v.
If u+v Z(F) and w+z Z(F) then u and v are contained by a path uwv of lengtp
If u+v Z(F) and w+z Z(F) then u and v are connected by a path uv of length=2.
Hence proved.
IV. REFERENCE

I. Beck, coloring of commutative ring, J. Algebra, 116(1) (1988),208226.

D.F. Anderson and P.S. Livingston, the zerodivisor graph of commutative ring, J. Algebra, 217(2) (1999), 434447

A.R. Ashrafi and A. Tadayyonfar, The zerodivisor graph of 2Ã—2 matrices over a field, vol.39., (2016), 977990
