Structure of intersection graphs

Let G be a finite group and let N be a fixed normal subgroup of G. In this paper, a new kind of graph on G, namely the intersection graph is defined and studied. We use Γ G (N) to denote this graph, with its vertices are all normal subgroups of G and two distinct vertices are adjacent if their intersection in N . We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of Γ G ({e}) for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups.


Introduction
Recently, study algebraic structures by graphs associated with them gives rise to many interesting results. Therefore, many mathematicians could associate the group theory with graph theory such as [1] and [4]. It has been proved that graphs can be interesting tools for the study of groups.
In the following context, some basics and related works are provided. Let H be a group. A non empty subset S of H is called subgroup if S is a group and denoted by S ≤ H. A subgroup S of H is called normal if h −1 sh ∈ S for all h ∈ H and s ∈ S [2]. A normal subgroup N of H is a minimal normal subgroup of H if 1 and N are the only normal subgroups of H that are contained in N . A finite group is solvable if all its composition factors are cyclic of prime order. www.ijc.or.id Let F q be a finite field. Then the general linear group GL(n, F q ) is the group of invertible n by n matrices with entries in F q under matrix multiplication. We define the special linear group, SL(n, F q ) = {A ∈ GL(n, F q ) : |A| = 1}.
A graph is connected if there is a path connecting any two distinct vertices. The distance between two distinct vertices is the length of the shortest path connecting them (if such a path does not exist, define ∞. The diameter of a graph G, denoted by diam(G), is defined by the supremum of the distances between vertices. The girth of a graph, denoted g(G) is the length of the shortest cycle in the graph G. A graph with no cycles has infinite girth. The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the graph G and denoted by rad(G).
The r-partite graph is one whose vertex can be partitioned into r subsets so that an edge has both ends in no subset. A complete r-partite graph is an r-partite graph in which each vertex is adjacent to every vertex that is not in the same subset. The complete bipartite graph with part sizes m and n is denoted by K m,n . A graph is called a complete if each pair of vertices is joined by an edge. We use K n to denote the complete graph with n vertices. Two graphs G and H are isomorphic, denoted by G ∼ = H, if there is a bijection φ : G → H of vertices such that the vertices x and y are adjacent in G if and only if φ(x) and φ(y) are adjacent in H. A connected graph can be drawn without any edges crossing, it is called planar. A vertex v of a connected graph G is called a cut vertex of G, if G \ v (Delete v from G) results in a disconnected graph. Removing a cut vertex from a graph breaks it into two or more graphs. A connected graph with no cycles is called a tree. [3]. Throughout this paper we consider a finite simple un-directed graph.

Theorem 1.2 (Kuratowski's Theorem). [3]
A graph is non-planar if and only if it contains a subgraph homeomorphic to K 3,3 or K 5 .

Some properties of the intersection graphs
In this section, we study the intersection graph of finite groups G. The structure of G is determined. As well, some related results are obtained.
The proof is clear.
Proof. First, it is clear that they have the same number of vertices. Let e = N i N j be an edge in On the other hand, let e = N i N j be an edge in Γ in G (G), that is N i ∩ N j ⊆ G. If either N i or N j is N r , then we are done. If neither N i nor N j is N r , then N i ∩ N j = Nr ⊆ N r wherer = min{i, j}. Thus e is an edge of Γ in G (G). Second, Since N i ∩ N j = N l ⊂ {e} for i, j ∈ {1, 2, ..., r + 1} where l = min{i, j} and N i ∩ {e} = {e}, then the result follows. Third, the proof is clear.
As a direct consequence of Proposition 2.1, the following results are obtained.
Proposition 2.2. If G is a finite solvable group and Γ in G ({e}) = K 1,2 , then 1. G is a cyclic p-group of order p 2 . 2. G is a semidirect product G = P Q, where P is an elementary abelian p-group and Q is a cyclic group of order q, with p and q being distinct primes.
Proof. Since Γ in G ({e}) = K 1,2 , then G has a unique non trivial normal subgroup. The proof of rests follow from Theorem 1.2 in [5].
Note that the converse of 1. in Proposition 2.2 is true and it can be seen in Proposition 3.3. The following example shows that the converse of 2. in Proposition 2.2 is not true.
The proof is clear.
Proposition 2.4. Let G be a finite group and N be a normal subgroup of G. Then Γ in G (N ) is connected graph with diameter at most 2 and radius 1.
Proof. The proof is clear. Proposition 2.5. Let G be a finite group with at least two minimal normal subgroups. Then Γ in G ({e}) is not tree graph. Furthermore, Γ in G (N ) has girth 3.
Proof. Since G has at least two minimal normal subgroups N 1 and N 2 , thus the normal subgroups with trivial normal subgroup gives the cycle Hence Γ in G (N ) has girth 3. Theorem 2.1. Let G be a finite group with non trivial normal proper subgroups N 1 , ..., N r . If |E(Γ in G (N i ))| = |E(Γ in G (N j ))| and |N i | = |N j | for some i, j. Then Γ in G (N i ) and Γ in G (N j ) are isomorphic.
Proof. The following examples show that the converse of Theorem 2.1 is not true.