The Intersection Graph of Subgroups of the Dihedral Group of Order

For a finite group G, the intersection graph of G is the graph whose vertex set is the set of all proper non-trivial subgroups of G, where two distinct vertices are adjacent if their intersection is a non-trivial subgroup of G. In this article, we investigate the detour index, eccentric connectivity, and total eccentricity polynomials of the intersection graph of subgroups of the dihedral group for distinct primes . We also find the mean distance of the graph .


Introduction
The concept of intersection graph of subgroups of a finite group was defined and studied by Csa'ka'ny and Polla'k in 1969 [1]. They found the clique number and degree of vertices of an intersection graph of subgroups of a dihedral group, quaternion group, and quasi-dihedral group. Let G be a finite non-abelian group. The intersection graph of is an undirected simple (without loops and multiple edges) graph whose vertex-set consists of all nontrivial proper subgroups of for which two distinct vertices H and K of are adjacent if ⋂ is a nontrivial subgroup of . This kind of graph has been studied by researchers; we refer the reader to see [2][3][4][5][6]. Let be any graph. The set of vertices and the set of edges of will be denoted by and , respectively. If there is an edge between vertices and , then we write . The cardinality of , denoted by | , is called the order of , while the cardinality of , denoted by | , is called the size of . For any vertex in the number of edges incident to is called the degree of and denoted by [7]. The chromatic number of a graph is , which is the smallest number of colors for such that adjacent vertices have different colors. A path is a walk with no two vertices repeated, for any two distinct vertices and in The shortest path in is called the distance between and denoted by and the longest path in is called the detour distance between , denoted by The eccentricity of a vertex denoted by is the longest distance between and all other vertices of The diameter of a graph , denoted by diam( , is defined as { The detour index, eccentric connectivity and total eccentricity polynomials are defined by ∑ [9], ∑ and ∑ [10], respectively. The detour index the eccentric connectivity index and the total eccentricity of a graph are the first derivatives of their corresponding polynomials at respectively. The transmission of a vertex in is ∑ The transmission of a graph is ∑ The mean (average) distance of graph is where is the order of [3,1,12]. Khasraw [13] studied the intersection graph of subgroups of the group , where , is a prime. He found some topological indices of the graph as well as its metric dimension and resolving polynomial.
In this paper, we consider the graph of the dihedral group where and are distinct primes. Some properties of the connected graph will be presented. The dihedral group of order 2pq is defined by 〈 〉 for prime numbers .

Some properties of the intersection graph of for prime numbers
In order to determine the vertex set of the graph , it is required to list all non-trivial proper subgroups of the dihedral group for distinct primes . In [6], the set of all non-trivial proper subgroups of the group are classified for all . Here, we only consider the case when for distinct primes . Lemma 2.1 [6].  [7].

Detour index, eccentric connectivity, and tot al eccentricity polynomials of the graph
In this section, we find detour index, eccentric connectivity, and total eccentricity polynomials of the intersection graph of .

The mean distance of the intersection graph
In this section, we find the mean distance of the intersection graph of subgroups of for distinct prime numbers and . Proof: Since the order of the graph is and the transmission of the graph is given in Theorem 4.1, we can find the mean distance of the graph as , where are prime numbers.