On Intersection Graph of Dihedral Group

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is the identity of the group $G$. In this paper, we investigate some properties and exploring some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric connectivity indices of the intersection graph of $D_{2n}$ for $n=p^2$, $p$ is prime. We also find the metric dimension and the resolving polynomial of the intersection graph of $D_{2p^2}$.


Introduction
The notion of intersection graph of a finite group has been introduced by Csákány and Pollák in 1969 [1]. For a finite group G, associate a graph Γ(G) with it in such away that the set of vertices of Γ(G) is the set of all proper non-trivial subgroups of G and join two vertices if their intersection is non-trivial. For more studies about intersection graphs of subgroups, we refer the reader to see [9,2,3,6,7].
Suppose that Γ is a simple graph, which is undirected and contains no multiple edges or loops. We denote the set of vertices of Γ by V (Γ) and the set of edges of Γ by E(Γ). We write uv ∈ E(Γ) if u and v form an edge in Γ. The size of the vertex-set of Γ is denoted by |V (Γ)| and the number of edges of Γ is denoted by |E(Γ)|. The degree of a vertex v in Γ, denoted by deg(v), is defined as the number of edges incident to v. The distance between any pair of vertices u and v in Γ, denoted by d (u, v), is the shortest u − v path in Γ. For a vertex v in Γ, the eccentricity of v, denoted by ecc(v), is the largest distance between v and any other vertex in Γ. The diameter of Γ, denoted as diam(Γ), is defined by diam(Γ) = max{ecc(v) : v ∈ V (Γ)}. A graph Γ is called complete if every pair of vertices in Γ are adjacent. If S ⊆ V (Γ) and no two elements of S are adjacent, then S is called an independent set. The cardinality of the largest independent set is called an independent number of the graph Γ. A graph Γ is called bipartite if the set V (Γ) can be partitioned into two disjoint independent sets such that each edge in Γ has its ends in different independent sets. A graph Γ is called split if V (Γ) can be partitioned into two different sets U and K such that U is an independent set and the subgraph induced by K is a complete graph. Let ). If distinct vertices have distinct representations with respect to W , then W is called a resolving set for Γ. A basis of Γ is a minimum resolving set for Γ and the cardinality of a basis of Γ is called the metric dimension of Γ and denoted by β(Γ) [8]. Suppose r i is the number of resolving sets for Γ of cardinality i. Then the resolving polynomial of a graph Γ of order n, denoted by β(Γ, x), is defined as β(Γ, x) = n i=β(Γ) r i x i . The sequence (r β(Γ) , r β(Γ)+1 , · · · , r n ) formed from the coefficients of β(Γ, x) is called the resolving sequence.
In [6], Rajkumar and Devi studied the intersection graph of subgroups of some non-abelian groups, especially the dihedral group D 2n , quaternion group Q n and quasi-dihedral group QD 2 α . They were only able to obtain the clique number and degree of vertices. It seems difficult to study most properties of the intersection graph of subgroups of these groups. In this paper, the focus will be on the intersection graph of subgroups of the dihedral group D 2n for the case when n = p 2 , p is prime. It is clear that when n = p, then the resulting intersection graph of subgroups is a null graph, which is not of our interest. For n = p 2 , the intersection graph Γ(D 2p 2 ) of the group D 2p 2 has p 2 +p+2 vertices. We leave the other possibilities for n open and we might be able to work on them in the future. So, all throughout this paper, the considered dihedral group is of order 2p 2 , and by intersection graph we mean intersection graph of subgroups.
This paper is organized as follows. In Section 2, some basic properties of the intersection graph of D 2p 2 are presented. We see that the intersection graph Γ(D 2p 2 ) is split. In Section 3, we find some topological indices of the intersection graph Γ(D 2p 2 ) of D 2p 2 such as the Wiener, hyper-Wiener and Zagreb indices. In Section 4, we find the metric dimension and the resolving polynomial of the intersection graph Γ(D 2p 2 ).

Some properties of the intersection graph of D 2n
In [6], all proper non-trivial subgroups of the group D 2n has been classified as shown in the following lemma. 2. cyclic groups H s = sr i of order 2, where i = 1, 2, · · · , n, and 3. dihedral groups H r s = r n k , sr i of order 2k, where k is a divisor of n, k = 1, n and i = 1, 2, · · · , n k . The total number of these proper subgroups is τ (n) + σ(n) − 2, where τ (n) is the number of positive divisors of n and σ(n) is the sum of positive divisors of n. We mentioned that we only focus on the case when n = p 2 , p is prime. Recall that, for n = p 2 , the intersection graph Γ(D 2p 2 ) of the group D 2p 2 has The following lemma is given in [6] to compute the degree of any vertex in Γ(D 2n ). Since we only consider the case n = p 2 , we restate it as follows: The following theorem gives the exact number of edges in Γ(D 2p 2 ) which can be in the Section 3 to compute the second Zagreb index.
. Proof. It follows from Theorem 2.2 that there are p 2 vertices of degree 1, p vertices of degree 2p + 1 and 2 vertices of degree p + 1. Thus, From Theorem 2.4, one can see that the maximum distance between any pair of vertices in Γ(D 2p 2 ) is 3. In order to explore the exact distance between any pair of vertices in Γ(D 2p 2 ) , we state the following corollary which can be used in the next section to find some topological indices of Γ(D 2p 2 ).
where i, j = 1, 2, ..., p. Proof. From Corollary 2.5, d(u, v) = 2 for every distinct pairs of vertices u, v ∈ H i and so uv / ∈ E(Γ). Therefore, H i is an independent set for each i.
Proof. From Theorem 2.6, there are p independent sets of size p. Also, from Corollary 2.5, one can see that none of the vertices of H 1,p is adjacent to vertices in H i for each i. So, in total the size of the largest independent set is p 2 + 1. The complete graph in the previous theorem is the largest complete subgraph of Γ(D 2n ). As a consequence, the clique number of Γ(D 2n ) is p + 2 which coincides with Theorem 2.3 in [6]. As a consequence of the above theorem, we have the following corollary.

Theorem 2.8. Let H ⊆ V (Γ(D 2p 2 )). Then the intersection graph Γ(H) is complete if and only if
Proof. It follows from Corollary 2.5 that no vertex of H 1,p ∪ {H i p } is adjacent to any vertex of H j , where i, j = 1, 2, ..., p and i = j. Then the maximum distance between any vertex of H 1,p ∪{H i p } and any other vertex in H j , i = j, is 2. Thus, ecc(v) = 2 for each v ∈ H 1,p ∪ {H i p }. Again, from Corollary 2.5, the maximum distance between any vertex of H i and any other vertex of H j , i = j, is 3, so ecc(v) = 3 for each v ∈ H i .
In the next two theorems, the first and second Zagreb indices for the intersection graph Γ(D 2n ) are presented.