Some Topological Indices of Subgroup Graph of Symmetric Group

The concept of the topological index of a graph is increasingly diverse because researchers continue to introduce new concepts of topological indices. Researches on the topological indices of a graph which initially only examines graphs related to chemical structures begin to examine graphs in general. On the other hand, the concept of graphs obtained from an algebraic structure is also increasingly being introduced. Thus, studying the topological indices of a graph obtained from an algebraic structure such as a group is very interesting to do. One concept of graph obtained from a group is subgroup graph introduced by Anderson et al in 2012 and there is no research on the topology index of the subgroup graph of the symmetric group until now. This article examines several topological indices of the subgroup graphs of the symmetric group for trivial normal subgroups. This article focuses on determining the formulae of various Zagreb indices such as first and second Zagreb indices and co-indices, reduced second Zagreb index and first and second multiplicatively Zagreb indices and several eccentricity-based topological indices such as first and second Zagreb eccentricity indices, eccentric connectivity, connective eccentricity, eccentric distance sum and adjacent eccentric distance sum indices of these graphs.


Introduction
The topological index of a finite graph is a number associated with the graph and this number is invariant under automorphism [1]. Topological index sometimes called a graph-theoretical descriptor [2][3][4] or molecular structure descriptor [5] of a graph. Various topological indices have been used to solve problem in biology and chemistry. Three major classifications of the topological index of a graph are based on degree, distance and the eccentricity of vertex in the graph.
Research on the topological index was initially related to graphs of biological activity or chemical structures and reactivity and researches in this regard continue, for example see [59][60][61][62][63]. On the other hand, several studies began to examine the topological index of graphs that are not of chemical structure and reactivity or biological activity, for example [54,[64][65][66][67][68][69][70][71][72]. When several researchers introduced new concepts about graphs obtained from an algebraic structure, research on topological indices on these graphs began to emerge, for example [73].
One concept of graphs obtained from a group is the concept of subgroup graph that was introduced by Anderson,Fasteen and LaGrange [74]. Referring to the definition of the subgroup graph by Anderson et al. [74], let G is a group and H is a normal subgroup in G. The subgroup graph Γ ( ) of group G is a simple and undirected graph with all elements of as its vertices and ∈ �Γ ( )� whenever ∈ for , ∈ and u ≠ v. As a result, the complement Γ ( ) �������� of the subgroup graph Γ ( ) is also simple and undirected [75]. Several studies related to graphs from a group have been widely reported, mostly of dihedral group [73, [76][77][78][79][80][81][82] and are still rare from symmetric groups [82][83][84]. Therefore, this article will examine the formulae of various Zagreb indices and eccentricity-based topological indices of the subgroup graph of the symmetric group.

Materials and Methods
All graphs in the present article is finite, simple, undirected and connected. For graph G = (V(G), E(G)), the order of G is ( ) = | ( )| and its size is ( ) = | ( )|. Let deg(u) denoted the degree of a vertex u in G. If deg(u) = 0, then u is an isolated vertex. If deg(u) = 1, then u is an end-vertex. Let ( , ) denoted the distance of vertex u and vertex v in G. The eccentricity e(u) of a vertex u is denotes the complete graph of order p. Then, ���� = 1 .
Zagreb index is the second oldest of degree-based topological index and Randic index is the first oldest. Firstly, the definition of various Zagreb index of graph G that will be used in this article are presented. After that, the definitions of several eccentricity-based topological indices which will also be used in this article are presented.
The following definitions refer to a graph G = (V(G), E(G)).
The first and second Zagreb indices of G are [17] and The first and second Zagreb co-indices of G are [11] and The reduced second Zagreb index of G is [12,14] The first and second multiplicative Zagreb indices of G are [13,15] and The first and second Zagreb eccentricity indices of G are [16] and The eccentric connectivity index of G is [86] The connective eccentricity index of G is [63] The adjacent eccentric distance sum index of G is [5]

Main Result
For a positive integer n ≥ 3, the symmetric group contains all permutations on the set ℕ = {1, 2, 3, … , } under the composition function operation. The symmetric group is a non-commutative group of order !. For distinct elements 1 , 2 , … , (k ≤ n) in ℕ , a cycle ( 1 2 ⋯ ) states a permutation π in such that ( ) = +1 and ( ) = 1 and π maps any other element of ℕ to itself. The cycle ( 1 2 ⋯ ) is called k-cycle or cycle with the length k. The 2-cycle is called transposition. The order of a k-cycle is k. The order of a product of disjoint transpositions has order 2. The order of permutation ρ in is 1 if and only if ρ = (1), where (1) is identity element of .
Throughout this paper, let X is the set of permutations in with order less than 3 and let Y is the set of permutations in with order more than 2. Hence, = ∪ .

if is even
Proof. The set X consists of identity permutation and permutations in term of products of disjoint 2-cycles. The number of permutations in with a given cycle structure is 100 Some Topological Indices of Subgroup Graph of Symmetric Group where denotes the number of k-cycles. By (14), the number permutation in the term of products of disjoint 2-cycles is This proves the theorem. ♦
Proof. Let δ is any element in Y. Then δ is a permutation of order more than 2. It implies that δ -1 is also a permutation of order more than 2. Hence, Y consists of permutations together with their inverses. This proves that | | is even. By Theorem 3.1,
The subgroup graph of a group will be simple and undirected if the subgroup is a normal subgroup. For symmetric group , the trivial normal subgroup is {(1)} and . This article only considers these two trivial normal subgroups of .

Theorem 3.5.
Proof. By definition of the subgroup graph, the edge set

The number of isolated-vertices in
Proof. From the proof of Theorem 3.5. ♦ According to Corollary 3.6., then Γ {1} ( ) can be written as with size as presented in (16). Therefore, the complement of Γ {1} ( ) is Thus, the following two theorems are obtained.

Theorem 3.8.
In As the direct consequences of Theorem 3.8., the following two corollaries are obtained.  presented as the following. a.
Some Topological Indices of Subgroup Graph of Symmetric Group

Conclusions
This article has presented the formulae of some degree-based and eccentric-based topological indices of the subgroup graphs of the symmetric group. The discussion in this article limited on trivial normal subgroups of symmetric group. For further research, examining on non-trivial normal subgroup needed to be studied.