1.
INTRODUCTION:
The phenomenon of
representing Groups using Graphs has been studied theoretically by number
of researchers [6,7,8,13,14,15]. In a series of investigations, S. Akbari and
A. Mohammadian [1] discussed the zero divisor graphs of finite rings. In [17],
maximum finite groups are represented as graphs with examples, diagrams
and theorems. In this paper, we give order prime graphs of finite groups
(multiplicative group of integers modulo n) and 
(Klein’s four group).
2.
ORDER PRIME GRAPH OF A GROUP:
2.1. Definition: Let Γ be a finite group. The order prime graph OP(Γ)
of a group Γ is a graph with V (OP(Γ))
= Γ and two vertices a and b are adjacent in OP(Γ) if and only if (o(a), o(b)) = 1.
Here o(a), o(b) respectively denote the orders of a and b .
2.2.
Order Prime Graph of 
The multiplicative
group of integers modulo 
is given by 
.The order prime
graphs of for some are given as follows:
2.2.1. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is complete graph.
(iii) This graph is connected graph
(iv)
The chromatic number is 
() = 1.
2.2.2. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is Bipartite because
the vertex set V can be decomposed into two disjoint subsets V1 and
V2 such that every edge in
has one end point in V1 and one end
point in V2.
(ii)
This graph is Regular because
every vertex is of same degree i.e., every vertex is of degree one.
(iii) This graph is finite because there are finite numbers
of vertices and edges.
(iv)
This graph is complete graph.
(v)
This graph is connected graph
(vi)
This graph is planar graph.
(vii) The chromatic number is () = 2.
2.2.3. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is connected graph
(iii) This graph is planar graph.
(iv)
The chromatic number is () = 2.
2.2.4. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is connected graph.
(iii)
The chromatic number is () =4.
2.2.5. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-This graph is finite graph.
(i)
This graph is connected graph.
(ii)
The chromatic number is () =3.
2.2.6. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is connected graph.
(iii)
This graph is planar graph.
(iv)
This graph is star graph.
(v)
The chromatic number is () =2.
2.2.7. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is connected graph.
(iii)
This graph is planar graph.
(iv)
This graph is star graph.
(v)
The chromatic number is () =2.
2.2.8. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is connected graph.
(iii)
The chromatic number is () =3.
2.2.9. Order Prime Graph of 
The multiplicative
group of integers modulo 
is 
The order prime graph of 
is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is connected graph.
(iii)
The chromatic number is () =3.
2.3. Order Prime Graph of 
The Klein’s four group is given by
, where 
and 
and 
The order prime graph of is:
Fig: 
We
have some following properties of this graph:-
(i)
This graph is finite graph.
(ii)
This graph is connected graph.
(iii)
This graph is planar graph.
(iv)
The chromatic number is () =2.
3. CONCLUSION:
In
this paper, we have drawn order prime graphs of 
(multiplicative group of integers modulo n) and 
(Klein’s four group) and found some properties of
these graphs. With the help of these graphs, we can find complement of order prime graphs of
these groups and their properties.
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and 