INTRODUCTION:

The phenomenon of representing Groups using Graphs has been studied theoretically by a number of researchers (See [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]). The interplay between groups and graphs have been the most famous and productive area of algebraic graph theory. In this paper, we give order prime graphs of Symmetric, Quaternion and Heisenberg groups.

2. ORDER PRIME GRAPH OF 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 Symmetric Group S_n:

2.2.1. Order Prime Graph of Symmetric Group S₂:

Let S₂={I, (12)} be the symmetric group of order 2. The order prime graph of S₂ is:

Fig. 1: OP(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 connected graph.

(v) This graph is complete graph.

(vi) This graph is star graph.

(vii) The chromatic number is (S₂) = 2.

2.2.2. Order Prime Graph of Symmetric Group S₃:

Let S₃={I, (12), (13), (23), (123), (132)} be the symmetric group of order 6. The order prime graph of S₃ is:

Fig. 2: OP(S₃)

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 (S₃) = 3.

2.2.3. Order Prime 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 prime graph of S₄ is:

Fig. 3: OP(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 Prime Graph of Quaternion Group Q₈:

Let Q₈be Quaternion group. The order prime graph of Q₈ is:

Fig. 4: OP(Q₈)

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 (Q₈) = 2.

2.4. Order Prime Graph of Heisenberg Group over Z_p:

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 Z_p.

2.4.2. Order Prime Graph of Heisenberg Group over Z₂:

Let

be Heisenberg Group of order 2³. The order prime graph of G is:

Fig. 5: OP(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.

2.4.3. Order Prime Graph of Heisenberg Group over Z₃:

Let

be Heisenberg Group of order 3³. The order prime graph of G is:

Fig. 6: OP(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.

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Received on 21.04.2018 Modified on 29.06.2018

Research J. Science and Tech. 2019; 11(1):38-42.

DOI: 10.5958/2349-2988.2019.00005.6