Some Topological Invariants of Graphs Associated with the Group of Symmetries

A topological index is a quantity that is somehow calculated from a graph (molecular structure), which reﬂects relevant structural features of the underlying molecule. It is, in fact, a numerical value associated with the chemical constitution for the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. A large number of properties like physicochemical properties, thermodynamic properties, chemical activity


Introduction
In quantitative structure-activity relationship (QSAR)/ quantitative structure-property relationship (QSPR) study, physicochemical properties and topological indices such as Randić index, atom-bond connectivity (ABC) index, and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numerical number. It correlates certain physicochemical properties such as boiling point, stability, and strain energy of chemical compounds of a molecular structure (graph). It is a numeric quantity associated with a chemical structure (graph), which characterizes the topology of the structure and is invariant under a structure-preserving mapping [1]. In 1947, Wiener [2] introduced the concept of (distancebased) topological index while working on the boiling point of paraffin. He named this index as the path number. Later on, the path number was renamed as the Wiener index [2], and then, the theory of topological indices started. Nowadays, a number of distance-based and degree-based topological indices have been introduced and computed (see for example [3][4][5][6][7][8][9][10][11][12][13][14][15], and the references therein).
We consider simple and connected graph (chemical structure) G with vertex set V(G) and edge set E(G). We denote the two adjacent vertices u and v in G as u ∼ v and nonadjacent vertices as u ≁ v. e number d v denotes the degree of a vertex v ∈ V(G) and e number d (u, v) denotes the length of a geodesic between u and v in G and is called the distance between u and v. e eccentricity of a vertex v in G, denoted by ecc(v), is the maximum distance between v and any other vertex of G. e minimum eccentricity amongst the vertices of G is called the radius of G, denoted by rad(G). e diameter of a graph G is the maximum eccentricity in G, denoted by D(G). A vertex v in G is said to be a central vertex if ecc(v) � rad(G), and the subgraph of G induced by central vertices of G is called the center of G. A vertex v in a graph G is called a peripheral vertex if ecc(v) � D(G), and the subgraph of G induced by peripheral vertices is called the periphery of G. e sum of two graphs G 1 and G 2 , denoted by G 1 + G 2 , is a graph with vertex set V(G 1 ) ∪ V(G 2 ) and an edge set E(G 1 Let Γ be a group. e set ζ(Γ) � x: x ∈ Γ ∧ xy � yx∀y ∈ Γ is called the center of the group Γ. en, commuting and noncommuting graphs of Γ are defined as follows: (i) e commuting graph of a nonabelian group Γ is denoted by Γ G � C(Γ, Ω) with vertex set Ω ⊆ Γ. For two distinct elements x, y ∈ Ω, x ∼ y in Γ G (x and y form an edge in Γ G ) if and only if xy � yx in Γ. e concept of commuting graphs on noncentral elements of a group has been studied by various researchers (see [16,17]). (ii) e noncommuting graph of a nonabelian group G Γ is a graph with vertex set V(G 1 ) ∪ V(G 2 ), and two distinct vertices u and v in G Γ form an edge if uv ≠ vu in Γ. e study of noncommuting graphs of groups was initiated in 1975 by Neumann [18] who posed the problem regarding the clique number of a noncommuting graph. Noncommuting graphs on noncentral elements of a group have also been studied by various other researchers [19,20]. e following useful property for a noncommuting graph was proposed in [19].
Proposition 1 (see [19]). For any nonabelian group Γ, Since ecc(v) ≤ 2 for every v ∈ G Γ , so we have the following straightforward proposition: Otherwise, it is the sum of the center and the periphery of G Γ .
is paper aimed at investigating all the topological properties (listed in Table 1) of commuting and noncommuting graphs associated with the group of symmetries. e rest of the paper consists of five sections. In the next section, we illustrate the group of symmetries and associate commuting and noncommuting graphs to this group. In Section 3, some useful constructions to investigate our main results of Sections 4 and 5 are provided.

Group of Symmetries and Associated Graphs
Group of symmetries finds its remarkable use in the theory of electron structures and molecular vibrations. Due to their significant employment in chemical structures, in the context of topological indices, we consider the group of symmetries of a regular polygon (also called a regular n-gon for n ≥ 2) in this paper. A regular n-gon is a geometrical figure all of whose sides have the same length and all the angles are of equal measurement. Each internal angle of a n-gon is of π − (2π)/(n) radian. e group of symmetries of a n-gon consists of 2n elements, which are n rotations (r 0 � e, r 1 , r 2 , . . . , r n− 1 about its center through an angle of (2kπ)/n radian, where k � 0, 1, . . . , n − 1, either all clockwise or all anticlockwise) and n reflections (for even n, the reflections through a line joining the midpoints of the opposite sides or through a line joining two opposite vertices, and for odd n, the reflections through those lines which join a vertex with the midpoint of the opposite side). e group of symmetries is denoted by D n and is called the dihedral group of order 2n. If we denote a rotation by "a" and a reflection by "b," then 2n elements of D n are a, a 2 , . . . , a n− 1 , a n � e and b, ab, a 2 b, . . . , a n− 1 b, where e is the identical rotation. e general representation of D n is given by  [21] RCW(G) [22] MTI(G) [23,24] H [11,12] R α (G) u∼v (d u × d v ) α Geometric arithmetic (GA) index [13] GA(G) s Fifth version of GA index [4] GA 5 [14] ABC(G) All the notations used in formulas are defined in Section 1.

Journal of Chemistry
with the center when n is odd, e, a n/2 , when n is even.
In the case of even value of n ≥ 4, we partitioned Ω 2 into n/2 two element subsets Remark 1. In the dihedral group D n , we have (i) xy � yx for all x, y ∈ D 2 (ii) a i b � ba n− i for i � 1, 2, . . . , n − 1 (iii) For odd values of n ≥ 3, xy ≠ yx for distinct x, y ∈ Ω 2 (iv) For even values of n ≥ 4, and for any distinct x, y ∈ Ω 2 , xy � yx if and only if x, y ∈ Ω i 2 , According to Remark 1, the commuting graph on D n is defined in the following result.
Proposition 3 (see [16]). For all n ≥ 3, let Γ G � C(D n , D n ) be a commuting graph on D n , then when n is odd, when n is even.
Here, K 1 is the trivial graph, K p is a complete graph on p vertices, N t is a null (empty) graph on t vertices, and (n/2)K 2 is the union of (n/2) copies of K 2 .
Let Γ � D n , n ≥ 3, and G Γ be the corresponding noncommuting graph. en, according to Remark 1, we have the following points: When n ≥ 3 is odd, then When n ≥ 4 is even, then and one partite set Ω 3 . Hence, by Proposition 2, we deduce the following result.
en, the noncommuting graph G Γ of D n is given by when n is odd,

Construction of Vertex and Edge Partitions
First we define some useful parameters, which support in the investigation of some predefined (in Table 1) topological indices. For any vertex v of G, these parameters are defined as follows: According to these parameters, the distance-based topological indices, listed in Table 1, become Let Γ G be a commuting graph of the dihedral group D n . In Γ G , there are 2n vertices. e number of edges in Γ G is (n(n + 1))/2 when n is odd and is (n(n + 4))/2 when n is even. Based on the degree, distance number, sum distance number, and reciprocal distance number of each vertex of Γ G , the useful vertex partition is given in Table 2. Based on degrees and degrees sum of the end vertices of each edge of Γ G , the useful edge partition is given in Table 3.
Let G Γ be a non-commuting graph of the dihedral group D n . In G Γ , (1) ere are 2n − 1 vertices and (3/2)n(n − 1) edges when n is odd (2) ere are 2n − 2 vertices and (3/2)n(n − 2) edges when n is even Based on the degree, eccentricity, distance number, sum distance number, and reciprocal distance number of each vertex of G Γ , the useful vertex partition is given in Table 4. Based on degrees and degrees sum of the end vertices of each edge of G Γ , the useful edge partition is given in Table 5.

Topological Properties of Commuting
Graph Γ G In this section, we compute the Wiener, reciprocal complementary Wiener, MTI, Harary general Randić, ABC, ABC 4 , GA, GA 5 , and harmonic indices of Γ G . roughout this section, in each of the two-row equation arrays, the first row corresponds to odd values of n, while the second corresponds to even values of n.
Theorem 1. For n ≥ 3, let Γ G be a commuting graph on D n , then Table 3: , when n is odd, n 2 (7n − 8), when n is even.
Proof. Using the vertex partition, given in Table 2, in formula (6) of the Wiener index, we have when n is odd, , when n is even.
Now, the required Wiener index can be obtained after some simplifications. (3n − 1), when n is odd, n 12 (21n − 16), when n is even.
Proof. Since the diameter of G Γ is 2, so by using the vertex partition, given in Table 4, in formula (7) of the reciprocal complimentary Wiener index, we have , when n is odd, when n is even.
Proof. Using the vertex partitions, given in Table 2, in formula (9) of the Harary index, we have
Also, for odd values of n, we have and for even values of n, we have e required formulas for both the indices one can get by performing an easy simplification.
Proof. By applying the formula of the harmonic index, using the edge partition given in Table 3, we have Some simplifications yield the required values of the harmonic index.
Proof. Since the diameter of G Γ is 2, so by sing the vertex partition, given in Table 4, in formula (7) of the reciprocal complimentary Wiener index, we have , when n is odd, , when n is even.
Exact values for this index are due to some easy calculations.
Proof. By sing the vertex partition, given in Table 4, in formula (9) of the Harary index, we have , when n is odd, , when n is even.
By performing some algebraic computations, one can obtain the required Harary index. □ Theorem 11. For n ≥ 3, let G Γ be a noncommuting graph of Γ � D n . en, for odd values of n, and for even values of n, Proof. Using the edge partition, given in Table 5, in the formula of general Randić index R α for α � 1, − 1, 1/2, − (1/2), we have , when n is odd, , when n is even, , when n is even, , when n is odd, , when n is even.
e required values of the geometric arithmetic index and its fifth version can be obtained after some simplifications. □ Theorem 13. For n ≥ 3, let G Γ be a noncommuting graph of Γ � D n , then when n is even, , when n is odd, when n is even.
Proof. By using the edge partition, given in Table 5, in formulas of ABC and ABC 4 indices, we have , when n is odd, , when n is even.
Proof. By applying the formula of the harmonic index, using the edge partition given in Table 5, we have , when n is odd, 2n(n − 2) 3n − 4 + n(n − 2) 2(2n − 4) , when n is even.
Some simplifications yield the required values of the harmonic index.

Concluding Remarks
An algebraic structure plays a vital role in chemistry to form chemical compound structures and in investigating various chemical properties of chemical compounds in these structures. Here, we considered a very well-known algebraic structure, called the group of symmetries of regular gons (the dihedral group), which has remarkable contribution in the theory of electron structures and molecular vibrations. We considered one algebraic property, namely, commutation property, on the dihedral group and associated two graphs (chemical structure) with the group of symmetries. We computed some distance-based and degree-based topological properties of these associated graphs by computing the exact formulae of the Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, Harary index, Randić index, geometric arithmetic indices, atomic bond connectivity indices, and harmonic index. All the indices are numeric quantities and, in fact, this work is a theoretical contribution in the theory of topological indices with the unique algebraic structure, and it can be very helpful to predict the bioactivity of chemical compounds using physicochemical properties in QSAR/QSPR studied.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.