Order Divisor Graphs of
Symmetric, Quaternion and Heisenberg Groups
Pankaj
Department of Mathematics,
Indira Gandhi University, Meerpur (Rewari)-122502, Haryana, India
*Corresponding Author E-mail: pankajarora1242@yahoo.com
Abstract:
We represent finite group in
the form of graphs. These graphs are called order divisor graphs. In this paper
we shall study order divisor graphs of Symmetric, Quaternion and Heisenberg
groups.
KEY WORDS: Symmetric group,
Quaternion group, Heisenberg group, Order Divisor graphs
2010 MATHEMATICS
SUBJECT CLASSIFICATIONS: 68R10, 97K30, 20B05, 20B30
1.
INTRODUCTION:
The phenomenon of
representing groups using graphs has been studied theoretically by a number
of researchers [1-11]. The interplay
between groups and graphs have been the most famous and productive area of
algebraic graph theory. In this article, we give order divisor graphs of
Symmetric, Quaternion and Heisenberg groups.
2.
ORDER DIVISOR GRAPH OF GROUP:
2.1.
Definition: We call a graph an order
divisor graph, denoted by OD(G), if its vertex set is a finite group G and two
distinct vertices a and b having different orders are adjacent, provided that
o(a) divides o(b) or o(b) divides o(a).
2.2. Order Divisor Graph of Symmetric
Group Sn:
2.2.1.
Order Divisor Graph of Symmetric Group S2: Let S2={I, (12)}
be the symmetric group of order 2. The order divisor graph of S2
is:
Fig. 1: OD(S2)
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 S2 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 
(S2)= 2.
2.2.2.
Order Divisor Graph of Symmetric Group S3: Let S3={I, (12),
(13), (23), (123), (132)} be the symmetric group of order 6. The order divisor
graph of S3 is:
Fig. 2: OD(S3)
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 (S3) = 2.
2.2.3.
Order Divisor Graph of Symmetric Group S4: Let S4={I, (12), (13), (14), (23),(24),
(34), (12)(34), (13)(24), (14)(23), (123), (132), (124), (142), (134), (143),
(234),(243), (1234), (1243), (1324), (1342), (1423), (1432)} be the symmetric
group of order 24. The order divisor graph of S4 is:
Fig. 3: OD(S4)
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 (S4) = 3.
2.3. Order Divisor Graph of Quaternion Group Q8:
Let Q8 be Quaternion group. The order
divisor graph of Q8 is:
Fig. 4: OD(Q8)
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 (Q8) = 3.
2.4. Order Divisor
Graph of Heisenberg Group over Zp:
2.4.1.
Definition: Let . Then 
is a non-abelian group of order p3
where 
is a field of order p. It is called Heisenberg
Group over Zp.
2.4.2.
Order Divisor Graph of Heisenberg Group over Z2:
Let
be Heisenberg
Group of order 23. The order divisor graph of G is:
Fig. 5: OD(G)
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 (G) = 3.
2.4.3.
Order Divisor Graph of Heisenberg Group over Z3:
2.4.4.
Let
be Heisenberg Group
of order 33. The order divisor graph of G is:
Fig. 6: OD(G)
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 (G) = 2.
3. CONCLUSION:
In this paper, we have drawn
order divisor graphs of Symmetric, Quaternion and Heisenberg groups
and found some properties of these graphs. With the help of these
graphs, we can find complement of order divisor graphs of these groups and
their properties.
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Received on 03.12.2017 Modified on 29.12.2017 Accepted on 10.01.2018 ©A&V Publications All right reserved Research J. Science and Tech. 2018; 10(1):68-72. DOI: 10.5958/2349-2988.2018.00009.8 |
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