# Some Topological Invariants of Graphs Associated with the Group of Symmetries

Authors: Chang-Cheng Wei; Muhammad Salman; Usman Ali; Masood Ur Rehman; Muhammad Aqeel Ahmad Khan; Muhammad Hasanain Chaudary; Farooq Ahmad
Journal: Journal of Chemistry
 (2020)

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## Abstract

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, and biological activity can be determined with the help of various topological indices such as atom-bond connectivity indices, Randić index, and geometric arithmetic indices. In this paper, we investigate topological properties of two graphs (commuting and noncommuting) associated with an algebraic structure by determining their Randić index, geometric arithmetic indices, atomic bond connectivity indices, harmonic index, Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, and Harary index.


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## Body

## 1. 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 (distance-based) 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 [3–15], and the references therein).We consider simple and connected graph (chemical structure)G with vertex set VG and edge set EG. We denote the two adjacent vertices u and v in G as u∼v and nonadjacent vertices as u≁v. The number dv denotes the degree of a vertex v∈VG and Sv=∑u∈Nvdu is the degrees sum of v, where Nv=u∈VG|u∼vinG is the neighborhood of v. The number du,v denotes the length of a geodesic between u and v in G and is called the distance between u and v. The eccentricity of a vertex v in G, denoted by eccv, is the maximum distance between v and any other vertex of G. The minimum eccentricity amongst the vertices of G is called the radius of G, denoted by radG. The diameter of a graph G is the maximum eccentricity in G, denoted by DG. A vertex v in G is said to be a central vertex if eccv=radG, 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 eccv=DG, and the subgraph of G induced by peripheral vertices is called the periphery of G. The sum of two graphs G1 and G2, denoted by G1+G2, is a graph with vertex set VG1∪VG2 and an edge set EG1∪EG2∪u∼v:u∈VG1∧v∈VG2.LetΓ be a group. The set ζΓ=x:x∈Γ∧xy=yx∀y∈Γ is called the center of the group Γ. Then, commuting and noncommuting graphs of Γ are defined as follows:(i)
The commuting graph of a nonabelian groupΓ is denoted by ΓG=CΓ,Ω with vertex set Ω⊆Γ. For two distinct elements x,y∈Ω, x∼y in ΓG (x and y form an edge in ΓG) if and only if xy=yx in Γ. The concept of commuting graphs on noncentral elements of a group has been studied by various researchers (see [16, 17]).(ii)
The noncommuting graph of a nonabelian groupGΓ is a graph with vertex set VG1∪VG2, and two distinct vertices u and v in GΓ form an edge if uv≠vu in Γ. The 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].The following useful property for a noncommuting graph was proposed in [19].Proposition 1 (see [19]).
For any nonabelian groupΓ, DGΓ=2.A graphG is said to be self-centered if radG=DG. Since eccv≤2 for every v∈GΓ, so we have the following straightforward proposition:Proposition 2.
A noncommuting graphGΓ of any nonabelian finite group Γ is self-centered if for each v∈GΓ, eccv=2. Otherwise, it is the sum of the center and the periphery of GΓ.This paper aimed at investigating all the topological properties (listed in Table1) of commuting and noncommuting graphs associated with the group of symmetries. The 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.Table 1
List of under consideration topological indices.
Name of the index
Notation
Formula
Wiener index [2]
W
G
∑
u
,
v
∈
V
G
d
u
,
v
Reciprocal complementary Wiener index [21]
RCW
G
∑
u
,
v
∈
V
G
1
/
D
G
+
1
−
d
u
,
v
Schultz molecular topological index [22]
MTI
G
∑
u
,
v
∈
V
G
d
u
+
d
v
d
u
,
v
+
∑
v
∈
V
G
d
2
/
v
Harary index [23, 24]
H
G
∑
u
≠
v
∈
V
G
1
/
d
u
,
v
Randić index [10]
R
−
1
/
2
G
∑
u
∼
v
1
/
d
u
×
d
v
General Randić index [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
G
∑
u
∼
v
2
S
u
×
S
v
/
S
u
+
S
v
Atomic bond connectivity (ABC) index [14]
ABC
G
∑
u
∼
v
d
u
+
d
v
−
2
/
d
u
×
d
v
Fourth version of ABC index [5]
ABC
4
G
∑
u
∼
v
S
u
+
S
v
−
2
/
S
u
×
S
v
Harmonic index [15]
H
r
G
∑
u
∼
v
2
/
d
u
+
d
v
All the notations used in formulas are defined in Section1.


## 2. 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 regularn-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. The group of symmetries of a n-gon consists of 2n elements, which are n rotations (r0=e,r1,r2,…,rn−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). The group of symmetries is denoted by Dn 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 Dn are a,a2,…,an−1,an=e and b,ab,a2b,…,an−1b, where e is the identical rotation. The general representation of Dn is given by(1)Dn=a,ban=b2=e,ab=ba−1,with the center(2)ζDn=e,whennis odd,e,an/2,whennis even.LetΩ1=e,a,a2,…,an−1, Ω2=b,ab,a2b,…,an−1b, and Ω3=Ω1−ζDn. Then Ω1=n=Ω2 and(3)Ω3=n−1,whennis odd,n−2,whennis even.In the case of even value ofn≥4, we partitioned Ω2 into n/2 two element subsets Ω2i=aib,an/2+ib, 0≤i≤n/2−1, so that Ω2=∩i=0n/2−1Ω2i.Remark 1.
In the dihedral groupDn, we have(i)
xy=yx for all x,y∈D2(ii)
aib=ban−i for i=1,2,…,n−1(iii)
For odd values ofn≥3, xy≠yx for distinct x,y∈Ω2(iv)
For even values ofn≥4, and for any distinct x,y∈Ω2, xy=yx if and only if x,y∈Ω2i, 0≤i≤n/2−1(v)
For any distinctx,y∈Ω3, xy=yx(vi)
For each pairx,y∈Ω2×Ω3, xy≠yx
According to Remark1, the commuting graph on Dn is defined in the following result.Proposition 3 (see [16]).
For alln≥3, let ΓG=CDn,Dn be a commuting graph on Dn, then(4)ΓG=K1+KΩ3∪NΩ1,whennis odd,K2+KΩ3∪n2K2,whennis even.Here,K1 is the trivial graph, Kp is a complete graph on p vertices, Nt is a null (empty) graph on t vertices, and n/2K2 is the union of n/2 copies of K2.LetΓ=Dn,n≥3, and GΓ be the corresponding noncommuting graph. Then, according to Remark 1, we have the following points:Whenn≥3 is odd, then(1)
Foru,v∈VGΓ,u∼v whenever u,v∈Ω2.(2)
Foru,v∈VGΓ,u∼v whenever u,v∈Ω3.(3)
Foru,v∈VGΓ,u∼v whenever u∈Ω2 and v∈Ω3.(4)
InGΓ, it can be seen that eccv=1 for all v∈Ω2, and eccv=2 for all v∈Ω3. It follows that Ω2 induces the center of GΓ, which is a complete graph KΩ2 on Ω2 vertices, and Ω3 induces the periphery of GΓ, which is a null graph NΩ3 on Ω3 vertices.Whenn≥4 is even, then(1)
Foru,v∈VGΓ,u≁v whenever u,v∈Ω2i for any 0≤i≤n/2−1.(2)
Foru,v∈VGΓ,u≁v whenever u,v∈Ω3.(3)
Foru,v∈VGΓ,u∼v whenever u∈Ω2 and v∈Ω3.(4)
Foru,v∈VGΓ,u∼v whenever u∈Ω2i and v∈Ω2j with 0≤i,j≤n/2−1 and i≠j.(5)
InGΓ, it can be seen that eccv=2 for all v∈Ω2∪Ω3. It follows that GΓ is a self-centered graph, which is a complete multipartite graph K2,2,…,2︸n/2−times,Ω3 with n/2 partite sets Ω2i,0≤i≤n/2−1, and one partite set Ω3.Hence, by Proposition2, we deduce the following result.Proposition 4.
Forn≥3, let Γ=Dn. Then, the noncommuting graph GΓ of Dn is given by(5)GΓ=KΩ2+NΩ3,whennis odd,K2,2,…,2︸n/2−times,Ω3,whennis even.


## 3. Construction of Vertex and Edge Partitions

First we define some useful parameters, which support in the investigation of some predefined (in Table1) topological indices. For any vertex v of G, these parameters are defined as follows:(i)
The distance number ofv in G is Dv|G=∑u∈VGdu,v(ii)
The sum distance number ofv in G is Dsv|G=∑u∈VG−v1/DG+1−du,v(iii)
The reciprocal distance number ofv in G is Drv|G=∑u∈VG1/du,vAccording to these parameters, the distance-based topological indices, listed in Table1, become(6)WG=12∑v∈VGDvG,(7)RCWG=GDG+1+12∑v∈VGDsvG,(8)MTIG=∑v∈VGdv2+∑v∈VGdvDvG,(9)HG=12∑v∈VGDrvG.LetΓG be a commuting graph of the dihedral group Dn. In ΓG, there are 2n vertices. The number of edges in ΓG is nn+1/2 when n is odd and is nn+4/2 when n is even. Based on the degree, distance number, sum distance number, and reciprocal distance number of each vertex of ΓG, the useful vertex partition is given in Table 2. Based on degrees and degrees sum of the end vertices of each edge of ΓG, the useful edge partition is given in Table 3.Table 2
Vertex partition ofΓG for each vertex v∈VΓG.
n is
d
v
D
v
|
Γ
G
D
s
v
|
Γ
G
D
r
v
|
Γ
G
Number of vertices
Odd
n
−
1
3
n
−
1
1
/
2
3
n
−
1
1
/
2
3
n
−
2
n
−
1
Odd
2
n
−
1
2
n
−
1
1
/
2
2
n
−
1
2
n
−
1
1
Odd
1
4
n
−
3
1
/
2
4
n
−
3
n
n
Even
n
−
1
3
n
−
1
1
/
2
3
n
−
1
1
/
2
3
n
−
2
n
−
2
Even
2
n
−
1
2
n
−
1
1
/
2
2
n
−
1
2
n
−
1
2
Even
3
4
n
−
5
1
/
2
4
n
−
5
n
+
1
nTable 3
Edge partition ofΓG for each edge u∼v∈EΓG.
n is
d
u
,
d
v type edges
S
u
,
S
v type edges
Number of edges
Odd
n
−
1
,
n
−
1
n
2
−
n
+
1
,
n
2
−
n
+
1
n
−
1
n
−
2
/
2
Odd
n
−
1,2
n
−
1
n
2
−
n
+
1
,
n
2
−
n
+
1
n
−
1
Odd
1,2
n
−
1
2
n
−
n
,
n
2
−
n
+
1
n
Even
n
−
1
,
n
−
1
n
2
+
1
,
n
2
+
1
n
−
2
n
−
3
/
2
Even
n
−
1,2
n
−
1
n
2
+
1
,
n
+
1
2
2
n
−
2
Even
2
n
−
1,2
n
−
1
n
+
1
2
,
n
+
1
2
1
Even
2
n
−
1,3
n
+
1
2
,
4
n
+
1
2
n
Even
3,3
4
n
+
1,4
n
+
1
n
/
2
Note:du,dv denotes the type of edge u∼v according to degrees of the end vertices, and Su,Sv denotes the type of edge u∼v according to degrees sum of the end vertices.LetGΓ be a non-commuting graph of the dihedral group Dn. In GΓ,(1)
There are2n−1 vertices and 3/2nn−1 edges when n is odd(2)
There are2n−2 vertices and 3/2nn−2 edges when n is evenBased on the degree, eccentricity, distance number, sum distance number, and reciprocal distance number of each vertex ofGΓ, the useful vertex partition is given in Table 4. Based on degrees and degrees sum of the end vertices of each edge of GΓ, the useful edge partition is given in Table 5.Table 4
Vertex partition ofGΓ for each vertex v∈VGΓ.
n is
d
v
ecc
v
D
v
|
G
Γ
D
s
v
|
G
Γ
D
r
v
|
G
Γ
Number of vertices
Odd
2
n
−
2
1
2
n
−
2
n
−
1
2
n
−
2
n
Odd
n
2
3
n
−
4
1
/
2
3
n
−
4
1
/
2
3
n
−
1
n
−
1
Even
2
n
−
2
2
2
n
−
2
n
−
1
1
/
2
4
n
−
7
n
Even
n
2
3
n
−
6
3
/
2
n
−
2
3
/
2
n
−
1
n
−
2Table 5
Edge partition ofGΓ for each edge u∼v∈EGΓ.
n is
d
u
,
d
v type edges
S
u
,
S
v type edges
Number of edges
Odd
n
,
2
n
−
2
2
n
n
−
1
,
n
−
1
3
n
−
2
n
n
−
1
Odd
2
n
−
2,2
n
−
2
n
−
1
3
n
−
2
,
n
−
1
3
n
−
2
n
n
−
1
/
2
Even
n
,
2
n
−
4
2
n
n
−
2
,
n
−
2
3
n
−
4
n
n
−
2
Even
2
n
−
4,2
n
−
4
n
−
2
3
n
−
4
,
n
−
2
3
n
−
4
n
n
−
2
/
2
Note:du,dv denotes the type of edge u∼v according to degrees of the end vertices, and Su,Sv denotes the type of edge u∼v according to degrees sum of the end vertices.


## 4. Topological Properties of Commuting GraphΓG

In this section, we compute the Wiener, reciprocal complementary Wiener, MTI, Harary general Randić,ABC, ABC4, GA, GA5, and harmonic indices of ΓG. Throughout 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.
Forn≥3, let ΓG be a commuting graph on Dn, then(10)WΓG=n27n−5,whennis odd,n27n−8,whennis even.Proof.
Using the vertex partition, given in Table2, in formula (6) of the Wiener index, we have(11)WΓG=n−13n−1+2n−1+n4n−32,whennis odd,n−23n−1+22n−1+n4n−52,whennis even.
Now, the required Wiener index can be obtained after some simplifications.Theorem 2.
Forn≥3, let ΓG be a commuting graph on Dn, then(12)RCWΓG=7n123n−1,whennis odd,n1221n−16,whennis even.Proof.
Since the diameter ofGΓ is 2, so by using the vertex partition, given in Table 4, in formula (7) of the reciprocal complimentary Wiener index, we have(13)RCWΓG=2n3+12n24n−3+122n−1+12n−13n−1,whennis odd,2n3+12n24n−5+2n−1+12n−23n−1,whennis even.
Now, the required index can be easily found by performing some simplifications.Theorem 3.
Forn≥3, let ΓG be a commuting graph on Dn, then(14)MTIΓG=2n2n−1n+1,whennis odd,2n2n−1n+4,whennis even.Proof.
By applying formula (8) of the Schultz molecular topological index using the vertex partition, given in Table 2, we have when n is odd(15)MTIΓG=n−13+2n−12+n+n−123n−1+2n−12+n4n−3=2n2n−1n+1,and when n is even(16)MTIΓG=n−2n−12+22n−12+n32+n−2n−13n−1+22n−12+3n4n−5=2n2n−1n+4.Theorem 4.
Forn≥3, let ΓG be a commuting graph on Dn, then(17)HΓG=n45n−1,whennis odd,n45n+2,whennis even.Proof.
Using the vertex partitions, given in Table2, in formula (9) of the Harary index, we have(18)HΓG=12n−13n−22+2n−1+n2,whennis odd,12n−23n−22+22n−1+nn+1,whennis even.
Some easy simplifications yield the required Harary index.Theorem 5.
Forn≥3, let ΓG be a commuting graph of Γ=Dn. Then, for odd values of n,(19)RαΓG=n2n3−n2+3n−1,forα=1,n4n−52n−22n−1,forα=−1,n−12n−2+2n−1n−12n−1+2n2n−12,forα=12,n−22n−1+2n−1+2n22n−1,forα=−12,and for even values of n,(20)RαΓG=12n−12n2−5n+6+22n−12n2+2n+3+9n,forα=1,92n−12n3−7n2+5n+224n3+20n2−11n+1818n−122n−12,forα=−1,n−1n−2n−3+7n−2+42n−1n−2n−1+n32,forα=12,2nn−12n−7+35n−43n−12n−1+2n−23+2nn−132n−1n−1,forα=−12.Proof.
Using the edge partition, given in Table3, in the formula of general Randić index Rα for α=1,−1,1/2,−1/2, we have(21)R1ΓG=n−13n−22+n−122n−1+n2n−1,n−2n−3n−122+2n−2n−12n−1+2n−12+6n2n−1+9n2,R−1ΓG=n−22n−1+12n−1+n2n−1,n−2n−32n−12+2n−2n−12n−1+12n−12+2n32n−1+n18,R1/2ΓG=n−1n−2n−122+n−1n−12n−1+n2n−1,n−2n−3n−12+2n−2n−12n−1+2n−1+2n32n−1+3n2,R−1/2ΓG=n−1n−22n−1n−1+n−1n−12n−1+n2n−1,n−2n−32n−1+2n−2n−12n−1+12n−1+2n32n−1+n6.
After a minor simplification, we get our required result.Theorem 6.
Forn≥3, let ΓG be a commuting graph of Γ=Dn, then(22)GAΓG=n−1n−22+2n−1n−12n−13n−2+2n−1,n2−4n+82+4n2−n−22n2−3n+1+2n3n−232n−1n+13n−2,GA5ΓG=nn2−1+4n3+n−132n+1,n2−4n+82+2n−2n+1n2+1n2+n+1+4nn+14n+1n2+6n+2.Proof.
Applying formulas of the geometric arithmetic index and its fifth version, using the edge partition given in Table3, we have(23)GAΓG=2n−12n−222n−2+2n−1n−12n−13n−2+2n2n−12n,n−2n−32+4n−2n−12n−13n−2+1+2n32n−1n+1+n2,GA5ΓG=2nn−1n2−n+124n2−n+1+2nn2−n+12n−1nn+1,n−2n−32+4n−2n2+1n+122n2+2n+2+1+4nn+124n+1n2+6n+2+n2.
The required values of the geometric arithmetic index and its fifth version can be obtained after some simplifications.Theorem 7.
Forn≥3, let ΓG be a commuting graph of Γ=Dn, then(24)ABCΓG=n−2n−22n−1+2n−13n−4+2nn−122n−1,n−2n−32n−22n−1+2n−12n−1+n3+2n2nn−1+2n−233n−43n−12n−1,ABC4ΓG=nn−12nn−12n2−n+1+nn2+n−22n−1n2−n+1,nn−2n−322n2+1+2n−2n+12nn+1n2+1+2nn+2n+12+2nn+1nn+64n+1+n2n4n+1.Proof.
By using the edge partition, given in Table3, in formulas of ABC and ABC4 indices, we have(25)ABCΓG=n−22n−22+n−13n−4n−12n−1+n2n−12n−1,n−2n−32n−22n−1+2n−23n−4n−12n−1+2n−12n−1+n3+2n2n32n−1.
Also, for odd values ofn, we have(26)ABC4ΓG=nn−12n2−n+1+n2−n+1−2n2−n+1n2−n+1+nn2−n+1+2n−1−2n2−n+12n−1,and for even values of n, we have(27)ABC4ΓG=n−2n−32n22n2+1+2n−2n+12n2+2nn2+1+2n2+4nn+12+2nn+1n2+6n4n+1+n8n24n+1.
The required formulas for both the indices one can get by performing an easy simplification.Theorem 8.
Forn≥3, let ΓG be a commuting graph of Γ=Dn, then(28)HrΓG=3n2+2n−423n−2,n−23n2−3n−22n−13n−2+6n+1+n2n−1n+136n+12n−1.Proof.
By applying the formula of the harmonic index, using the edge partition given in Table3, we have(29)HrΓG=n−1n−22n−1+2n−13n−2+1,n−2n−32n−1+4n−23n−2+12n−1+2nn+1+n6.
Some simplifications yield the required values of the harmonic index.


## 5. Topological Properties of Noncommuting GraphGΓ

In this section, we compute the reciprocal complementary Wiener, Harary, general Randić,ABC, ABC4, GA, GA5, and harmonic indices of GΓ.Theorem 9.
Forn≥3, let GΓ be a noncommuting graph of Γ=Dn, then(30)RCWGΓ=11215n2−19n+4,whennis odd,11215n2−34n+28,whennis even.Proof.
Since the diameter ofGΓ is 2, so by sing the vertex partition, given in Table 4, in formula (7) of the reciprocal complimentary Wiener index, we have(31)RCWGΓ=2n−13+nn−12+n−13n−44,whennis odd,2n−23+nn−12+3n−2n−24,whennis even.
Exact values for this index are due to some easy calculations.Theorem 10.
Forn≥3, let GΓ be a noncommuting graph of Γ=Dn, then(32)HGΓ=147n2−8n+1,whennis odd,147n2−16n+6,whennis even.Proof.
By sing the vertex partition, given in Table4, in formula (9) of the Harary index, we have(33)HGΓ=n2n−22+n−13n−14,whennis odd,n4n−74+3n−2n−14,whennis even.
By performing some algebraic computations, one can obtain the required Harary index.Theorem 11.
Forn≥3, let GΓ be a noncommuting graph of Γ=Dn. Then, for odd values of n,(34)RαGΓ=2nn−122n−1,forα=1,5n−48n−1,forα=−1,nn−12nn−1+n−1,forα=12,nn−12+n4,forα=−12,and for even values of n,(35)RαGΓ=4nn−22n−1,forα=1,10n−1616n−32,forα=−1,nn−2n2n−4+nn−22,forα=12,4nn−2+n2nn−242nn−2,forα=−12.Proof.
Using the edge partition, given in Table5, in the formula of general Randić index Rα for α=1,−1,1/2,−1/2, we have(36)R1GΓ=n2n−12n−2+nn−12n−222,whennis odd,n2n−22n−4+nn−22n−422,whennis even,R−1GΓ=nn−1n2n−2+nn−122n−22,whennis odd,nn−2n2n−4+nn−222n−42,whennis even,R1/2GΓ=nn−1n2n−2+nn−12n−22,whennis odd,nn−2n2n−4+nn−22n−42,whennis even,R−1/2GΓ=nn−1n2n−2+nn−122n−2,whennis odd,nn−2n2n−4+nn−222n−4,whennis even.
By performing some simplifications, we get the required results.Theorem 12.
LetGΓ be a noncommuting graph of Γ=Dn, for n≥3, then(37)GAGΓ=22nn−13/2+nn−13n−223n−2,whennis odd,22nn−23/2+nn−23n−423n−4,whennis even,GA5GΓ=4n2nn−123n−2+nn−15n−210n−4,whennis odd,nn−25n−4+32n3n−410n−8,whennis even.Proof.
Applying formulas of the geometric arithmetic index and its fifth version, using the edge partition given in Table5, we have(38)GAGΓ=2nn−1n2n−23n−2+nn−12,whennis odd,2nn−2n2n−43n−4+nn−22,whennis even,GA5GΓ=2nn−12nn−123n−22nn−1+n−13n−2+nn−12,whennis odd,2nn−22nn−223n−42nn−2+n−23n−4+nn−22,whennis even.
The required values of the geometric arithmetic index and its fifth version can be obtained after some simplifications.Theorem 13.
Forn≥3, let GΓ be a noncommuting graph of Γ=Dn, then(39)ABCGΓ=nn−13n−42+n22n−38,whennis odd,nn−23n−62+n4n−104,whennis even,ABC4GΓ=n2n−125n2−7n6n4−16n3+14n2−4n+n26n2−10n+223n−2,whennis odd,5n3−14n2+6n3n−4+2n2n−13n−723n−4,whennis even.Proof.
By using the edge partition, given in Table5, in formulas of ABC and ABC4 indices, we have(40)ABCGΓ=nn−13n−42nn−1+nn−14n−622n−2,whennis odd,nn−23n−6n2n−4+nn−24n−1022n−4,whennis even.
Also, for odd values ofn, we have(41)ABC4GΓ=nn−12nn−1+n−13n−2−22nn−123n−2+nn−122n−13n−2−2n−13n−22,and for even values of n, we have(42)ABC4GΓ=nn−22nn−2+n−23n−4−22nn−223n−4+nn−22n−23n−4−22n−23n−4.
The required formulas for both the indices one can get by performing an easy simplification.Theorem 14.
Forn≥3, let GΓ be a noncommuting graph of Γ=Dn, then(43)HrGΓ=n11n−1043n−2,whennis odd,n11n−2043n−4,whennis even.Proof.
By applying the formula of the harmonic index, using the edge partition given in Table5, we have(44)HrGΓ=2nn−13n−2+nn−122n−2,whennis odd,2nn−23n−4+nn−222n−4,whennis even.
Some simplifications yield the required values of the harmonic index.


## 6. 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.



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