Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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670 
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Zagreb Indices of an Undirected graphs  𝑮𝒏 and  𝑮𝒎,𝒏
𝑴

 

 

L.Eswaramma1, D.Venkata Lakshmi2*, M.Siva Parvathi3 

1Department of Mathematics, RGUKT Ongole campus, Andhra Pradesh, India. 

Email: eswaramma@rguktong.ac.in 
2*School of Computer Science and Engineering,VIT-AP University, Andhra Pradesh, India. 

*Corresponding Author: himaja96@gmail.com 
3Department of Applied Mathematics, Sri Padmavati MahilaVisvavidyalayam, Tirupathi, Andhra Pradesh, India. 

Email: parvathimani2008@gmail.com 

 

Article History: 

Received: 06-06-2024 

Revised: 05-07-2024 

Accepted: 30-07-2024 

Abstract 

A Topological index is a molecular structure descriptor of a molecular graph and it has many 

applications in real world issues. The Zagreb Indices based on degree of vertices of a graph are 

widely studied in chemical graph theory in over the past few years. Degree based Zagreb 

indices of undirected graphs  𝐺𝑚,𝑛
𝑀 and  𝐺𝑛 graphs are computed in this paper. 

Conclusions: The authors of this paper have studied the Zagreb indices of undirected       

graphs 𝐺𝑛 when   𝑛 = 2𝛼, 𝛼 > 2 and undirected graph  𝐺𝑚,𝑛
𝑀  for some cases that is when   

𝑛 = 2𝑝, 𝑝 is prime,𝑚 > 𝑛, 𝑚 is prime and when   𝑚 > 𝑛, 𝑚, 𝑛 are odd primes.      

Keywords: Zagreb indices of a graph, undirected graph   𝐺𝑚,𝑛
𝑀, undirected graph   𝐺𝑛. 

 

1. Introduction 

Topological index is a numerical quantity which plays vital role in QSAR or QSPR studies. Different 

types of topological indices are introduced and studied so far. In 1972 Gutman [1], [2] introduced 

Zagreb indices to describe the properties of chemical compounds, the chemical structures are 

described by molecular graphs. This formula in graph theory represents atoms and chemical bonds as 

vertices and edges, respectively. Some bounds for the Zagreb indices discussed in [3-5]. The 

topological indices of graph operations are computed in [8,9]. New types of Zagreb indices are 

presented in [10].Siva Parvathi et al [11] calculated the Zagreb indices of selected chemical 

compounds of natural products. 

 Ivy Chakrabarthy et al [12] introduced the undirected graph 𝐺𝑛 and proved some basic properties. 

Ivy Chakrabarthy et al [13] introduced the undirected graph  𝐺𝑚,𝑛
𝑀  and proved some basic 

properties of 𝐺𝑚,𝑛
𝑀 graph. Recently Eswaramma et al [14] calculated energy of this  𝐺𝑚,𝑛

𝑀 graph. 

Anusha et al[15]calculated the Zagreb indices of Arithmetic graphs. 

 Motivated by these, we calculate the Zagreb indices of undirected 𝐺𝑚,𝑛
𝑀graph, undirected 𝐺𝑛graph. 

Definitions 

Consider the simple graph G. Let 𝑑𝑖, 𝑑𝑗 be the degrees of the vertices 𝑣𝑖 , 𝑣𝑗  and 𝑒𝑖𝑗 be the edges 

joining the vertices 𝑣𝑖 , 𝑣𝑗  respectively.Then  

The First Zagreb index of graph G is defined as  

             𝑀1(𝐺) = 


+
)(,

)(
GEvv

ji

ji

dd

 



Communications on Applied Nonlinear Analysis 

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The Second Zagreb index of graph G is defined as      

           𝑀2(𝐺) = 



)(,

)(
GEvv

ji

ji

dd

 

The hyper-Zagreb index was introduced by Shirdel et al [6] in 2013.The hyper Zagreb index of graph G 

is defined as   

          𝑀𝐻(𝐺) =
2

)(,

)(


+
GEvv

ji

ji

dd  

In [7], M.Ghorbani and N.Azimi defined different versions of the Zagreb indices based on the degree 

of vertices. 

First multiple Zagreb index of the graph G is defined as 

         𝑃𝑀1(𝐺)= 


+
)(,

)(
GEvv

ji

ji

dd  

Second multiple Zagreb index of the graph G is defined as 

         𝑃𝑀2(𝐺)= 



)(,

)(
GEvv

ji

ji

dd

 

2. Zagreb indices of an undirected graph  𝑮𝒎,𝒏
𝑴

 

The Undirected simple graph 𝐺𝑚,𝑛
𝑀 is 𝐺𝑚,𝑛

𝑀 = (𝑉, 𝐸) where the vertex set  𝑉 = {1,2, … . 𝑛} and two 

distinct vertices  𝑢, 𝑣 ∈ 𝑉 are adjacent if and only if 𝑢 ≠ 𝑣and 𝑢. 𝑣 is not divisible by 𝑚 on natural 

numbers subset which is finite where 𝑚, 𝑛 ∈ 𝑁. Some of the properties are  

      1. Let 𝑚 = 1 then the graph 𝐺𝑚,𝑛
𝑀 is a null graph with 𝑛 vertices. 

      2. For 1 < 𝑚 ≤ 𝑛,the graph 𝐺𝑚,𝑛
𝑀 is disconnected. 

      3. The graph   𝐺𝑚,𝑛
𝑀is connected for 𝑚 > 𝑛 

     4. The graph 𝐺𝑚,𝑛
𝑀  has the Maximum degree   𝑛 − 1. 

         Results on various Zagreb indices of the graph 𝐺𝑚,𝑛
𝑀 are presented in this section. 

Theorem 2.1: If  𝐺𝑚,𝑛
𝑀   be an undirected graph where   𝑛 = 2𝑝, 𝑝 is prime,    𝑚 > 𝑛  , 𝑚 is prime. Then  

 (i) First Zagreb index 𝑀1( 𝐺𝑚,𝑛
𝑀 ) is 𝑛(𝑛 − 1)2. 

 (ii)Second Zagreb index 𝑀2( 𝐺𝑚,𝑛
𝑀 ) is  

𝑛(𝑛−1)3

2
. 

 (iii)Hyper Zagreb index 𝑀𝐻( 𝐺𝑚,𝑛
𝑀 ) is 2𝑛(𝑛 − 1)3. 

 ( iv)First multiple Zagreb index  𝑃𝑀1( 𝐺𝑚,𝑛
𝑀 ) is 2(𝑛 − 1)

𝑛(𝑛−1)

2 . 

  (iv) Second multiple Zagreb index  𝑃𝑀2( 𝐺𝑚,𝑛
𝑀 ) is (𝑛 − 1)𝑛(𝑛−1). 

Proof:  Consider an undirected graph  𝐺𝑚,𝑛
𝑀 where   𝑛 = 2𝑝, 𝑝 is prime, 𝑚 > 𝑛,𝑚 is prime with 𝑉 =

{1,2,3 … 𝑛} as the vertex set, the vertex degree is (𝑛 − 1) for every 𝑣 ∈ 𝑉and 𝐸 is the edge set. 

Let 𝑑𝑖 , 𝑑𝑗 be the degrees of the vertices 𝑣𝑖 , 𝑣𝑗  and 𝑒𝑖𝑗 be the edges joining the vertices   𝑣𝑖 , 𝑣𝑗 . 



Communications on Applied Nonlinear Analysis 

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Then     |𝑒𝑖𝑗| =
𝑛(𝑛−1)

2
  

(i)First Zagreb index of the graph  𝐺𝑚,𝑛
𝑀 is 

  𝑀1( 𝐺𝑚,𝑛
𝑀) = 



+
Evv

ji

ji

dd
,

)(  

                      = |𝑒𝑖𝑗|(𝑑𝑖 + 𝑑𝑗) 

                      = 
𝑛(𝑛−1)

2
[(𝑛 − 1) + (𝑛 − 1)] 

                      = 𝑛(𝑛 − 1)2. 

(ii)Second Zagreb index of the graph  𝐺𝑚,𝑛
𝑀  is 

 𝑀2( 𝐺𝑚,𝑛
𝑀

) = 



Evv

ji

ji

dd
,

)(   = |𝑒𝑖𝑗|(𝑑𝑖 × 𝑑𝑗) 

                     = 
𝑛(𝑛−1)

2
[(𝑛 − 1) × (𝑛 − 1)] 

                     = 
𝑛(𝑛−1)3

2
 

(iii) Hyper Zagreb index of the graph  𝐺𝑚,𝑛
𝑀  is 

 𝑀𝐻( 𝐺𝑚,𝑛
𝑀) =

2

)(,

)(


+
GEvv

ji

ji

dd  

                      = |𝑒𝑖𝑗|(𝑑𝑖 + 𝑑𝑗)2 

                     = 
𝑛(𝑛−1)

2
[(𝑛 − 1) + (𝑛 − 1)]2 

                     = 2𝑛(𝑛 − 1)3. 

(iv)First multiple Zagreb index of the graph  𝐺𝑚,𝑛
𝑀 is 

 𝑃𝑀1( 𝐺𝑚,𝑛
𝑀)= 



+
)(,

)(
GEvv

ji

ji

dd

 

                       = (𝑑𝑖 + 𝑑𝑗)|𝑒𝑖𝑗| 

                       = [(𝑛 − 1)+(𝑛 − 1)]
𝑛(𝑛−1)

2  

                       = (2(𝑛 − 1))
𝑛(𝑛−1)

2 . 

(v) Second multiple Zagreb index of the graph  𝐺𝑚,𝑛
𝑀  is 

𝑃𝑀2( 𝐺𝑚,𝑛
𝑀)= 




Evv

ji

ji

dd
,

)(

 

                      

= (𝑑𝑖 × 𝑑𝑗)|𝑒𝑖𝑗| 

                      = [(𝑛 − 1)(𝑛 − 1)]
𝑛(𝑛−1)

2  



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                      = (𝑛 − 1)𝑛(𝑛−1). 

 Theorem 2.2: If  𝐺𝑚,𝑛
𝑀   be an undirected graph where   𝑚 > 𝑛, 𝑚, 𝑛 are odd primes. Then  

(i) First Zagreb index 𝑀1( 𝐺𝑚,𝑛
𝑀 ) is 𝑛(𝑛 − 1)2. 

(ii)Second Zagreb index 𝑀2( 𝐺𝑚,𝑛
𝑀 ) is  

𝑛(𝑛−1)3

2
. 

(iii)Hyper Zagreb index 𝑀𝐻( 𝐺𝑚,𝑛
𝑀 ) is2𝑛(𝑛 − 1)3. 

 ( iv)First multiple Zagreb index  𝑃𝑀1( 𝐺𝑚,𝑛
𝑀 ) is 2(𝑛 − 1)

𝑛(𝑛−1)

2 . 

  (iv) Second multiple Zagreb index  𝑃𝑀2( 𝐺𝑚,𝑛
𝑀 ) is (𝑛 − 1)𝑛(𝑛−1). 

Proof:  Consider an undirected graph  𝐺𝑚,𝑛
𝑀 with  𝑚 > 𝑛, 𝑚, 𝑛 are odd primes and 𝑉 = {1,2,3 … 𝑛}is 

the vertex set .Here the degree of the vertex is (𝑛 − 1) for every 𝑣 ∈ 𝑉. 

Let 𝑑𝑖 , 𝑑𝑗 be the degrees of the vertices 𝑣𝑖 , 𝑣𝑗  and 𝑒𝑖𝑗 be the edges joining the vertices 𝑣𝑖 , 𝑣𝑗 . 

Then  |𝑒𝑖𝑗| =
𝑛(𝑛−1)

2
 . 

(i)First Zagreb index of the graph  𝐺𝑚,𝑛
𝑀 is 

 𝑀1( 𝐺𝑚,𝑛
𝑀) = 



+
Evv

ji

ji

dd
,

)(     

                     = |𝑒𝑖𝑗|(𝑑𝑖 + 𝑑𝑗) 

                     = 
𝑛(𝑛−1)

2
[(𝑛 − 1) + (𝑛 − 1)] 

                     = 𝑛(𝑛 − 1)2. 

(ii)Second Zagreb index of the graph  𝐺𝑚,𝑛
𝑀  is 

 𝑀2( 𝐺𝑚,𝑛
𝑀

) = 



Evv

ji

ji

dd
,

)(             

                     = |𝑒𝑖𝑗|(𝑑𝑖 × 𝑑𝑗) 

                     = 
𝑛(𝑛−1)

2
[(𝑛 − 1) × (𝑛 − 1)] 

                     = 
𝑛(𝑛−1)3

2
 

(iii) Hyper Zagreb index of the graph  𝐺𝑚,𝑛
𝑀  is  

   𝑀𝐻( 𝐺𝑚,𝑛
𝑀) =

2

,

)(


+
Evv

ji

ji

dd

 

                       

= |𝑒𝑖𝑗|(𝑑𝑖 + 𝑑𝑗)2 

                       = 
𝑛(𝑛−1)

2
[(𝑛 − 1) + (𝑛 − 1)]2 

                       = 2𝑛(𝑛 − 1)3. 

(iv)First multiple Zagreb index of the graph  𝐺𝑚,𝑛
𝑀 is 



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 𝑃𝑀1( 𝐺𝑚,𝑛
𝑀)= 



+
Evv

ji

ji

dd
,

)(

 

                      

= (𝑑𝑖 + 𝑑𝑗)|𝑒𝑖𝑗| 

                      = [(𝑛 − 1)+(𝑛 − 1)]
𝑛(𝑛−1)

2  

                      = (2(𝑛 − 1))
𝑛(𝑛−1)

2 . 

(v) Second multiple Zagreb index of the graph  𝐺𝑚,𝑛
𝑀  is 

𝑃𝑀2( 𝐺𝑚,𝑛
𝑀)= 




Evv

ji

ji

dd
,

)(    

                      = (𝑑𝑖 × 𝑑𝑗)|𝑒𝑖𝑗| 

                      = [(𝑛 − 1)(𝑛 − 1)]
𝑛(𝑛−1)

2  

                      = (𝑛 − 1)𝑛(𝑛−1). 

3. Zagreb Indices of the undirected graph  𝑮𝒏 

Let an undirected simple graph 𝐺𝑛 = (𝑉, 𝐸) whose vertex set 𝑉 is a subset of Natural numbers 

defined as 𝑉 = {𝑥 ∈ 𝑁/(𝑥, 𝑛) ≠ 1, 𝑥 < 𝑛} , where 𝑛 ∈ 𝑁and 𝑛 is not a prime number and a pair of 

vertices 𝑥, 𝑦 ∈ 𝑉 is adjacent if and only if gcd (𝑥, 𝑦) > 1.Some of the properties of graph  𝐺𝑛 as 

follows 

           1. The graph 𝐺𝑛 is complete if and only if  𝑛 = 𝑝𝑚 where 𝑝 is prime. 

           2. The graph 𝐺𝑛 is disconnected if and only if 𝑛 = 2𝑝 where 𝑝 is an odd prime. 

 Results on various Zagreb indices of the graph 𝐺𝑛  are presented in this section.  

Theorem 3.1: If  𝐺𝑛 be an undirected graph where   𝑛 = 2𝛼, > 2 . Then  

(i) First Zagreb index 𝑀1(𝐺𝑛) is   (2α−1 − 1)(2α−1 − 2)2. 

 (ii)Second Zagreb index 𝑀2(𝐺𝑛) is   
(2𝛼−1−1)(2𝛼−1−2)3

2
 . 

 (iii)Hyper Zagreb index 𝑀𝐻(𝐺𝑛) is 2(2𝛼−1 − 1)(2𝛼−1 − 2)3. 

 ( iv)First multiple Zagreb index  𝑃𝑀1(𝐺𝑛) is  (2(2𝛼−1 − 2))
(2𝛼−1−1)(2𝛼−1−2)

2 . 

  (iv) Second multiple Zagreb index  𝑃𝑀2(𝐺𝑛) is (2𝛼−1 − 2)(2𝛼−1−1)(2𝛼−1−2). 

Proof:  Consider an undirected graph 𝐺𝑛 where 𝑛 = 2𝛼, 𝛼 > 2  with the vertex set𝑉 =
{2,2.2,3.2, … . (2𝛼−1 − 1). 2}and 𝐸 is the edge set.Here the degree of the vertex is (2𝛼−1 − 2) for 

every   𝑣 ∈ 𝑉. 

Let 𝑑𝑖 , 𝑑𝑗 be the degrees of the vertices 𝑣𝑖 , 𝑣𝑗  and 𝑒𝑖𝑗 be the edges joining the vertices   𝑣𝑖 , 𝑣𝑗 . 

 Then    |𝑒𝑖𝑗| =
(2𝛼−1−1)(2𝛼−1−2)

2
  

(i)First Zagreb index of the graph 𝐺𝑛  where 𝑛 = 2𝛼, 𝛼 > 2 is 



Communications on Applied Nonlinear Analysis 

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   𝑀1(𝐺𝑛) = 


+
Evv

ji

ji

dd
,

)(  

                = |𝑒𝑖𝑗|(𝑑𝑖 + 𝑑𝑗) 

                = 
(2𝛼−1−1)(2𝛼−1−2)

2
 [(2𝛼−1 − 2) + (2𝛼−1 − 2)] 

                = (2α−1 − 1)(2α−1 − 2)2. 

(ii)Second Zagreb index of the graph 𝐺𝑛 where    𝑛 = 2𝛼, 𝛼 > 2 is 

 𝑀2(𝐺𝑛) = 



Evv

ji

ji

dd
,

)(  

              = |𝑒𝑖𝑗|(𝑑𝑖 × 𝑑𝑗) 

              =
(2𝛼−1−1)(2𝛼−1−2)

2
[(2𝛼−1 − 2) × (2𝛼−1 − 2)]  

              = 
(2𝛼−1−1)(2𝛼−1−2)3

2
 

(iii) Hyper Zagreb index of the graph 𝐺𝑛 where    𝑛 = 2𝛼, 𝛼 > 2  is  

 𝑀𝐻(𝐺𝑛) =
2

,

)(


+
Evv

ji

ji

dd   

               = |𝑒𝑖𝑗|(𝑑𝑖 + 𝑑𝑗)2 

               = 
(2𝛼−1−1)(2𝛼−1−2)

2
 [(2𝛼−1 − 2) + (2𝛼−1 − 2)]2 

               = 2(2𝛼−1 − 1)(2𝛼−1 − 2)3. 

(iv) First multiple Zagreb index of the graph 𝐺𝑛 where   𝑛 = 2𝛼, 𝛼 > 2  is 

 𝑃𝑀1(𝐺𝑛)= 


+
Evv

ji

ji

dd
,

)(

 

               = (𝑑𝑖 + 𝑑𝑗)|𝑒𝑖𝑗|                              

               =[(2𝛼−1 − 2) + (2𝛼−1 − 2)]
(2𝛼−1−1)(2𝛼−1−2)

2  

               = (2(2𝛼−1 − 2))
(2𝛼−1−1)(2𝛼−1−2)

2 . 

(v) Second multiple Zagreb index of the graph 𝐺𝑛 where   𝑛 = 2𝛼, 𝛼 > 2  is 

      𝑃𝑀2(𝐺𝑛) = 



Evv

ji

ji

dd
,

)(  

                      = (𝑑𝑖 × 𝑑𝑗)|𝑒𝑖𝑗| 

                      = [(2𝛼−1 − 2)(2𝛼−1 − 2)]
(2𝛼−1−1)(2𝛼−1−2)

2  

                      = (2𝛼−1 − 2)(2𝛼−1−1)(2𝛼−1−2). 



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676 
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4. Conclusion 

The authors of this paper have studied the Zagreb indices of undirected graphs 𝐺𝑛 when   𝑛 = 2𝛼, 𝛼 >

2 and undirected graph  𝐺𝑚,𝑛
𝑀  for some cases that is when   𝑛 = 2𝑝, 𝑝 is prime,𝑚 > 𝑛, 𝑚 is prime and 

when   𝑚 > 𝑛, 𝑚, 𝑛 are odd primes.      

5. Refrences 

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polynes.J.Chem.Phys.62,1975,pp.3399-3405. 

[3] K.C.Das,I.Gutman and B.Zhou.,”New upper bounds on Zagreb Indices”,Journal of Mathematical 

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[4] K.C.Das  A.Yurttas,M,Togan,A.S.Cevik,I.N.Cangul,”The Multiplicative Zagreb Indices of graph 

operations”,Journal of Inequalities and Applications ,vol.2013,Article ID 90,2013. 

[5] G.H.Fath –Tabar.,”Old and new Zagreb Indices of graphs”,MATCH Communications in Mathematical and in 

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[6] Shirdel GH,Rezapour H,AM sayadi-“The Hyper-Zagreb index of graph operations”, Iranian Journal of 

Mathematical Chemistry,4(2),pp.213-220,2013. 

[7] Ghorbani.M,Azimi A-“A note on multiplicative Zagreb indices” , Iranian Journal of Mathematical 

Chemistry,3(2),pp.137-143,2012. 

[8] Das.K.C.,Gutman.I, “Some properties of the second Zagreb Index”, MATCH Commun.Math.Comput.Chem.52, 

pp.103-112,2004. 

[9] A.R.Ashrafi,T.Doslic,A.Hamzeh,-“The Zagreb coindices of graph operations”,Discrete Appl,Math.158,1571-

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[10] Zhou.B.-Zagreb Indices , MATCH Commun.Math.Comput.Chem.52,pp.113-118,2004. 

[11] M.Siva Parvathi,D.Sujatha and T.Sukeerthi,“A Stastical Comparision between Zagreb Indices for correlation with 

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