The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(4) (2023), Pages 513-520 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: July 07, 2022 Reviewed: February 02, 2023 Accepted: February 25, 2023 
DOI: http://dx.doi.org/10.18860/ca.v7i4.16991  

The First Zagreb Index, The Wiener Index, and The Gutman Index 
of The Power of Dihedral Group 

Evi Yuniartika Asmarani1, Sahin Two Lestari1, Dara Purnamasari1, Abdul Gazir 
Syarifudin2, Salwa1, I Gede Adhitya Wisnu Wardhana1*  

1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University 
of Mataram, Indonesia 

2Department of Magister Mathematics, Faculty of Mathematics and Natural Sciences, 
Institut Teknologi Bandung, Indonesia. 

Email: adhitya.wardhana@unram.ac.id  

ABSTRACT  

Research on graphs combined with groups is an interesting topic in the field of combinatoric algebra where 
graphs are used to represent a group. One type of graph representation of a group is a power graph. A power 
graph of the group 𝐺 is defined as a graph whose vertex set is all elements of 𝐺 and two distinct vertices 𝑎 
and 𝑏 are adjacent if and only one vertice is the power of other vertice. In addition to mathematics, graph 
theory can be applied to various fields of science, one of which is chemistry, which is related to topological 
indices. In this study, the topological indexes will be discussed, namely the Zagreb index, the Wiener index, 
and the Gutman index of the power graph of the dihedral group  where 𝑛 is a prime power. The method used 
in this research is a literature review.  For the main result, we gives the first Zagreb index, Wiener index, and 
Gutman index of the power graph of the dihedral group. 

Copyright © 2023 by Authors, Published by CAUCHY Group. This is an open access article under the CC BY-
SA License (https://creativecommons.org/licenses/by-sa/4.0/) 
 
Keywords: first Zagreb index; Wiener index; Gutman index; power graph; dihedral group 

INTRODUCTION 

In mathematics, graph theory has many uses, especially in algebraic structures 
where graphs are used to represent a group. Many types of graphs are developed from a 
group, one of which is a power graph. The first power graph introduced by Kalarev in 
2013 [1] is to define a directed power graph of a semigroup. And motivated by this, Askin 
and Buyukkose discuss the undirected power graph of semigroups and groups [2]. 
Recently, there have been many studies discussing the power graph of a group, one of 
which is the study by Asmarani et al. which deals with the power graph of a dihedral group 
when 𝑛 = 𝑝𝑚 where 𝑝 is a prime number and an 𝑚 is a natural number [3]–[5]. 

Besides mathematics, graph theory has benefits in other fields, one of which is 
chemistry, which is related to topological indices. Topological indices represent chemical 
structures and are useful for predicting the chemical and physical properties of molecular 
structures. Not only researching graphs related to chemical structures but over time 
research on topological indices has developed to examine graphs in general. Several types 

http://dx.doi.org/10.18860/ca.v7i4.16991
mailto:adhitya.wardhana@unram.ac.id
https://creativecommons.org/licenses/by-sa/4.0/


The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group 

Evi Yuniartika Asmarani 514 

of topological indexes are interesting to discuss. Some of them are the first Zagreb index, 
the Wiener index, and the Gutman index [6]. 

Research on the topological index of a graph, especially graphs related to groups is 
interesting to do. Several previous research results discuss the topological index of graphs 
related to groups, namely the topological index of the non-commuting graph of a dihedral 
group, the Szeged and Wiener indices for the coprime graph of the dihedral group [7], and 
connectivity indices of the coprime graph of generalized quaternion group [8].  For other 
graph representations of a group, see [8]–[14].  Recently, not many studies have 
investigated the topological indices of graphs associated with groups, especially the 
power graphs of dihedral groups. Therefore, in this study, topological indices will be 
discussed, namely the first Zagreb index, Wiener index, and the Gutman index of the 
power graph of a dihedral group when 𝑛 = 𝑝𝑚  where 𝑝 is a prime number and an 𝑚 is a 
natural number. 

 

METHODS 

This study uses a deductive proof method to find new knowledge from an algebraic 
structure from a previous study.  We start by studying the algebraic structure for several 
cases looking for a pattern.  And with the foundation of the pattern, we stated the 
conjecture for a general case, if the conjecture is proven by deductive proof, the conjecture 
is stated as a theorem.   

RESULTS AND DISCUSSION  

Preliminaries 

In this section, we present some definitions and theorems that are needed in this 
research. 
Definition 1 [15] Group 𝐺 is said to be a dihedral group of order 2𝑛, 𝑛 ≥ 3, and 𝑛 ∈ ℕ, is 
a group composed of two elements 𝑎,𝑏 ∈ 𝐺 with the property 

𝐺 = 〈𝑎,𝑏|𝑎𝑛 = 𝑒,𝑏2 = 𝑒,𝑏𝑎𝑏−1 = 𝑎−1〉 

 The dihedral group of order 2n is denoted by 𝐷2𝑛.   

Definition 2 [5] Power graph of group G denoted by 𝒢(𝐺) is an undirected graph whose 
vertex set is G and two vertices 𝑎,𝑏 ∈ 𝐺 are adjacent if and only if 𝑎 ≠ 𝑏 and 𝑎𝑚 = 𝑏 or 
𝑏𝑚 = 𝑎 for some positive integer 𝑚. 

 
We will give some topological indices of the power graph such as the Zagreb index, 

Wiener index, and Gutman index.  The definitions are as follows 
 
Definition 3 [16] Let 𝒢 be a simple connected graph. The first Zagreb index of 𝒢, 
denoted by 𝑀1(𝒢), is defined as  

𝑀1(𝒢) = ∑ (deg(𝑣))
2

𝑣∈𝑉(𝒢)

 

where 𝑑𝑒𝑔(𝑣) is the number of edges that incident to 𝑣. 
 
Definition 4 [2] Let 𝒢 be a simple connected graph. The Wiener index of 𝒢, denoted by 
𝑊(𝒢), is defined as  



The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group 

Evi Yuniartika Asmarani 515 

𝑊(𝒢) = ∑ 𝑑(𝑢, 𝑣)

𝑢,𝑣∈𝑉(𝒢)

 

where 𝑑(𝑢,𝑣) is the shortest distance between vertex 𝑢 and 𝑣. 
 
Definition 5 [17] Let 𝒢 be a simple connected graph. The Gutman index of 𝒢, denoted by 
𝐺𝑢𝑡(𝒢), is defined as  

𝐺𝑢𝑡(𝒢) = ∑ deg(𝑢)deg(𝑣)𝑑(𝑢,𝑣)

𝑢,𝑣∈𝑉(𝒢)

 

where 𝑑𝑒𝑔(𝑢) and deg⁡(𝑣) are the number of edges that incident to 𝑢 and 𝑣 and 𝑑(𝑢,𝑣) 
are the shortest distance between vertex 𝑢 and 𝑣. 
 
Theorem 1 [3] If 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers, then the 
power graph of a dihedral group is a graph that has two non-disjoint subgraphs, namely 
a complete subgraph and a star subgraph. 
 
Example 1 Power graph of the dihedral group 𝐷2.3 as shown in the following figure 

 

FIGURE 1. Power graph of the dihedral group 𝐷2.3 

Theorem 2 [3] The vertex degree of the power graph of a dihedral group when 𝑛 = 𝑝𝑚 

where 𝑝 prime numbers and an 𝑚 natural numbers are 

a. deg(𝑒) = 2𝑛 − 1 

b. deg(𝑎𝑖) = 𝑛 − 1 for every 𝑖 ∈ ℤ,1 ≤ 𝑖 ≤ 𝑛 − 1 

c. deg(𝑎𝑗𝑏) = 1 for every 𝑗 ∈ ℤ, 0 ≤ 𝑗 ≤ 𝑛 − 1 

Main Result 

 If 𝑛 = 𝑝𝑚 where 𝑝 is prime and an 𝑚 natural number then the first Zagreb index, 
Wiener index, and Gutman index of the power graph of a dihedral group, respectively 

is 𝑛2(𝑛 + 1),
7𝑛2

2
−

5𝑛

2
,
1

2
⁡(𝑛4 + 𝑛) +

3

2
(𝑛3 − 𝑛2) as shown in the following theorem. 

Theorem 3 If 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers, then the Zagreb 
index of the power graph of the dihedral group 𝐷2𝑛 is 𝑛

2(𝑛 − 1). 
Proof. 

𝑀1(𝒢(𝐷2𝑛))⁡⁡⁡⁡⁡⁡⁡⁡= ∑ deg(𝑢)
2

𝑢∈𝑉(𝒢(𝐷2𝑛))

 



The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group 

Evi Yuniartika Asmarani 516 

= deg(𝑒)2 .1 + ∑ deg(𝑎𝑖)
2

𝑛−1

𝑖=1

+ ∑ deg(𝑎𝑖𝑏)
2

𝑛−1

𝑖=0

 

= (2𝑛 − 1)2. 1 + (𝑛 − 1)2(𝑛 − 1) + 12.𝑛 
= 4𝑛2 − 4𝑛 + 1 + (𝑛 − 1)3 + 𝑛 
= 4𝑛2 − 4𝑛 + 1 + 𝑛3 − 3𝑛2 + 3𝑛 − 1 + 𝑛 
= 𝑛3 + 𝑛2 
= 𝑛2(𝑛 + 1) 

Theorem 4 If 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers then the Wiener 

index of the power graph of the dihedral group 𝐷2𝑛 is 
7𝑛2

2
−

5𝑛

2
. 

 
Proof. Let 𝐷2𝑛 = {𝑒, 𝑎,𝑎

2,…,𝑎𝑛−1,𝑏,𝑎𝑏,…,𝑎𝑛−1𝑏}, a dihedral group with 𝑛 = 𝑝𝑚 where 
𝑝 is a number prime and 𝑚 natural number then the dihedral group can be partitioned 
into 3 partitions namely 𝑉1 = {𝑒}, 𝑉2 = {𝑎,𝑎

2,…,𝑎𝑛−1⁡} and 𝑉3 = {𝑏,𝑎𝑏,𝑎
2⁡𝑏, …, 𝑎𝑛−1𝑏}. 

To prove the Wiener index of the power graph of a dihedral group can be divided into 4 
cases. 
Case 1. 
For 𝑒 ∈ 𝑉1 and 𝑥 ∈ 𝑉(𝒢(𝐷2𝑛) where 𝑒 ≠ 𝑣, obtained 

∑ 𝑑(𝑒,𝑥)

𝑥∈𝐷2𝑛
∗

= (2𝑛 − 1).1 

              = 2𝑛 − 1 
 
Case 2 
For 𝑎𝑖,𝑎𝑗 ∈ 𝑉2 where  1 ≤ 𝑖 ≤ 𝑛 − 1, 1 ≤ 𝑗 ≤ 𝑛 − 1, and 𝑖 ≠ 𝑗 obtained 

∑ 𝑑(𝑎𝑖,𝑎𝑗)⁡= (
𝑛 − 1

2
).1

1≤𝑖≤𝑛−1
⁡1≤𝑗≤𝑛−1

𝑖≠𝑗

 

=
(𝑛 − 1)!

2!(𝑛 − 3)!
 

=
(𝑛 − 1)(𝑛 − 2)

2
 

=
𝑛2

2
−
3𝑛

2
+ 1 

 
Case 3  
For 𝑎𝑐𝑏,𝑎𝑑𝑏 ∈ 𝑉3 where 0 ≤ 𝑐 ≤ 𝑛 − 1, 0 ≤ 𝑑 ≤ 𝑛 − 1, and 𝑐 ≠ 𝑑 obtained 

∑ 𝑑(𝑎𝑐𝑏, 𝑎𝑑𝑏) = (
𝑛

2
).2

0≤𝑐≤𝑛−1
0≤𝑑≤𝑛−1

𝑐≠𝑑

 

=
𝑛!

2! (𝑛 − 2)!
.2 

= 𝑛(𝑛 − 1) 
= 𝑛2 − 𝑛 

  
Case 4 
For 𝑎𝑒 ∈ 𝑉2 and 𝑎

𝑓𝑏 ∈ 𝑉3 where 1 ≤ 𝑒 ≤ 𝑛 − 1, 0 ≤ 𝑓 ≤ 𝑛 − 1 obtained 



The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group 

Evi Yuniartika Asmarani 517 

∑ 𝑑(𝑎𝑒,𝑎𝑓𝑏)⁡⁡⁡⁡⁡⁡= (𝑛 − 1)𝑛.2
1≤𝑒≤𝑛−1
0≤𝑓≤𝑛−1

 

= 2𝑛2 − 2𝑛  
Based on definition 4 and the four cases, the Wiener index of the power graph of the 
dihedral group 𝐷2𝑛 when 𝑛 = 𝑝

𝑚 for a prime number 𝑝 and 𝑚 natural numbers is: 

𝑊((𝒢(𝐷2𝑛) = ∑ 𝑑(𝑢, 𝑣)

𝑢,𝑣∈𝑉(𝒢(𝐷2𝑛)

 

=⁡ ∑ 𝑑(𝑒,𝑥)

𝑥∈𝐷2𝑛
∗

+ ∑ 𝑑(𝑎𝑖,𝑎𝑗)
1≤𝑖≤𝑛−1
⁡1≤𝑗≤𝑛−1

𝑖≠𝑗

+⁡ ∑ 𝑑(𝑎𝑐𝑏,𝑎𝑑𝑏)
0≤𝑐≤𝑛−1
0≤𝑑≤𝑛−1

𝑐≠𝑑

⁡

+ ∑ 𝑑(𝑎𝑒,𝑎𝑓𝑏)
1≤𝑒≤𝑛−1
0≤𝑓≤𝑛−1

 

= (2𝑛 − 1) + (
𝑛2

2
−
3𝑛

2
+ 1) + (𝑛2 − 𝑛) + (2𝑛2 − 2𝑛) 

=
7𝑛2

2
−
5𝑛

2
 

Teorema 5 If 𝑛 = 𝑝𝑚 with 𝑝 prime numbers and an 𝑚 natural numbers then the Gutman 

index of the power graph of the dihedral group 𝐷2𝑛 is 
1

2
(𝑛4 + 𝑛) +

3

2
(𝑛3 − 𝑛2). 

Proof. Let 𝐷2𝑛 = {𝑒, 𝑎,𝑎
2,…,𝑎𝑛−1,𝑏,𝑎𝑏,…,𝑎𝑛−1𝑏}, a dihedral group with 𝑛 = 𝑝𝑚 where 

𝑝 is a number prime and an 𝑚 natural number then the dihedral group can be partitioned 
into 3 partitions namely 𝑉1 = {𝑒}, 𝑉2 = {𝑎,𝑎

2,…,𝑎𝑛−1⁡} and 𝑉3 = {𝑏,𝑎𝑏,𝑎
2⁡𝑏, …, 𝑎𝑛−1𝑏}. 

To prove the Gutman index of the power graph of a dihedral group can be divided into 5 
cases. 
Case 1 
For 𝑒 ∈ 𝑉1 and 𝑎

𝑖 ∈ 𝑉2 where 1 ≤ 𝑖 ≤ 𝑛 − 1, obtained 

∑ (deg(𝑒) .deg(𝑎𝑖)).𝑑(𝑒,𝑎𝑖) = (((2𝑛 − 1)(𝑛 − 1))1)(𝑛 − 1)

1≤𝑖≤𝑛−1

 

         = 2𝑛3 − 5𝑛2 + 4𝑛 − 1  
Case 2 
For 𝑒 ∈ 𝑉1 and 𝑎

𝑗𝑏 ∈ 𝑉3 where 0 ≤ 𝑗 ≤ 𝑛 − 1, obtained 

∑ (deg(𝑒).deg(𝑎𝑗𝑏)).𝑑(𝑒,𝑎𝑗𝑏) = (((2𝑛 − 1)1)1)𝑛

0≤𝑗≤𝑛−1

= 2𝑛2 − 𝑛 

Case 3 
For 𝑎𝑘, 𝑎𝑙 ⁡∈ 𝑉2 where 1 ≤ 𝑘 ≤ 𝑛 − 1, 1 ≤ 𝑙 ≤ 𝑛 − 1, and 𝑘 ≠ 𝑙, obtained 

∑ (deg(𝑎𝑘) .deg(𝑎𝑙)).𝑑(𝑎𝑘,𝑎𝑙) = (((𝑛 − 1)(𝑛 − 1))1)
1≤𝑘≤𝑛−1
1≤𝑙≤𝑛−1

𝑘≠𝑙

(
𝑛 − 1

2
) 

= (𝑛 − 1)(𝑛 − 1)
(𝑛 − 1)!

2!(𝑛 − 3)!
 

=
(𝑛 − 1)3(𝑛 − 2)

2
 

=
𝑛4

2
−
5𝑛3

2
+
9𝑛2

2
−
7𝑛

2
+ 1 

Case 4 



The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group 

Evi Yuniartika Asmarani 518 

For 𝑎𝑐𝑏,𝑎𝑑𝑏⁡ ∈ 𝑉3 where 0 ≤ 𝑐 ≤ 𝑛 − 1,⁡0 ≤ 𝑑 ≤ 𝑛 − 1, and 𝑐 ≠ 𝑑, obtained 

∑ (deg(𝑎𝑐𝑏).deg(𝑎𝑑𝑏)).𝑑(𝑎𝑐𝑏,𝑎𝑑𝑏) = (1.1.2)
⁡0≤𝑐≤𝑛−1
0≤𝑑≤𝑛−1

𝑐≠𝑑

(
𝑛

2
) 

= 2(
𝑛

2
) 

= 2
𝑛!

2! (𝑛 − 2)!
 

= 𝑛(𝑛 − 1) 
= 𝑛2 − 𝑛 

Case 5  
For 𝑎𝑒 ∈ 𝑉2 and 𝑎

𝑓𝑏 ∈ 𝑉3 where 1 ≤ 𝑒 ≤ 𝑛 − 1,⁡0 ≤ 𝑓 ≤ 𝑛 − 1, obtained 

∑ (deg(𝑎𝑒). deg(𝑎𝑓𝑏)).𝑑(𝑎𝑒,𝑎𝑓𝑏)⁡= (((𝑛 − 1)1)2)
1≤𝑒≤𝑛−1
0≤𝑓≤𝑛−1

(𝑛 − 1)𝑛 

= 2𝑛(𝑛 − 1)(𝑛 − 1) 
= 2𝑛(𝑛2 − 2𝑛 + 1) 
= 2𝑛3 − 4𝑛2 + 2𝑛 

Based on definition 5 and the five cases, the Gutman index of the power graph of the 
dihedral group 𝐷2𝑛 when 𝑛 = 𝑝

𝑚 for a prime number p and m natural numbers is: 

𝐺𝑢𝑡(𝒢(𝐷2𝑛)) = ∑ (deg(𝑢).deg(𝑣)).𝑑(𝑢,𝑣)

𝑢,𝑣∈𝑉(𝒢(𝐷2𝑛)

 

= ∑ (deg(𝑒). deg(𝑎𝑖)).𝑑(𝑒,𝑎𝑖)

1≤𝑖≤𝑛−1

+ ∑ (deg(𝑒).deg(𝑎𝑗𝑏)).𝑑(𝑒,𝑎𝑗𝑏)⁡

0≤𝑗≤𝑛−1

+ ∑ (deg(𝑎𝑘) .deg(𝑎𝑙)).𝑑(𝑎𝑘,𝑎𝑙)
1≤𝑘≤𝑛−1
1≤𝑙≤𝑛−1

𝑘≠𝑙

+ ∑ (deg(𝑎𝑐𝑏). deg(𝑎𝑑𝑏)).𝑑(𝑎𝑐𝑏,𝑎𝑑𝑏)
⁡0≤𝑐≤𝑛−1
0≤𝑑≤𝑛−1

𝑐≠𝑑

+ ∑ (deg(𝑎𝑒) .deg(𝑎𝑓𝑏)).𝑑(𝑎𝑒,𝑎𝑓𝑏)
1≤𝑒≤𝑛−1
0≤𝑓≤𝑛−1

= (2𝑛3 − 5𝑛2 + 4𝑛 − 1) + (2𝑛2 − 𝑛) + (
𝑛4

2
−
5𝑛3

2
+
9𝑛2

2
−
7𝑛

2
+ 1) + (𝑛2

− 𝑛) + (2𝑛3 − 4𝑛2 + 2𝑛) 

=
𝑛4

2
+
3𝑛3

2
−
3𝑛2

2
+
𝑛

2
 

=
1

2
(𝑛4 + 𝑛) +

3

2
(𝑛3 − 𝑛2) 



The First Zagreb Index, The Wiener Index, and The Gutman Index of The Power of Dihedral Group 

Evi Yuniartika Asmarani 519 

CONCLUSIONS 

The results obtained from this study are the first Zagreb index, Wiener index, and Gutman 
index of the power graph for the dihedral group 𝐷2𝑛 where 𝑛 = 𝑝

𝑚, 𝑝 is prime and 𝑚 a 

natural number respectively is 𝑛2(𝑛 + 1),
7𝑛2

2
−

5𝑛

2
,  
1

2
 (𝑛4 + 𝑛) +

3

2
(𝑛3 − 𝑛2). 

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