Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

254 
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Topological Indices of ARMS-Product of Certain Graphs 

 

Roy John.a 1, Akhil B.b 2 and Manju V. N. b 3  
a Department of Mathematics, St. Stephen’s College, Pathanapuram, Kollam, Kerala, India. Pin:689695. 

 b Department of Mathematics, University of Kerala, Karyavattom, Thiruvananthapuram, Kerala, India. Pin:695581. 

a 1Corresponding Author: roymaruthoor@gmail.com  

b 2 akhilb@keralauniversity.ac.in  b 3 manjushaijulal@gmail.com 

 

Article History: 

Received: 01-08-2024 

Revised: 09-09-2024 

Accepted: 18-09-2024 

Abstract:  

A mathematical parameter known as the topological graph index, or molecular descriptor, 

can be applied to any graph that represents a molecule structure. This index can be used to 

analyze mathematical numbers and look into specific physical and chemical characteristics 

of molecules in more detail. It is therefore a useful strategy for avoiding costly and time-

consuming laboratory experiments. In this paper topological indices of a noval graph 

product called ARMS-Product of certain graphs are determined. 

Keywords: ARMS-Product, first Zagreb index, second Zagreb index, Harmonic index, 

Geometric-arithmetic index, Bull graph, Path. 

2020 AMS subject classifications: 05C07, 05C76, 05C92. 

1. Introduction 

Topological indices are graph invariant. A graph invariant is any function on a graph that does not 

depend on labelling of its vertices. In chemistry, graphs can represent different chemical objects such 

as molecules, crystals, polymers, etc. Molecular graphs are chemical graphs which represent the 

constitution of molecules. In these graphs, individual atoms represent vertices and edges are chemical 

bonds between them. A single number that can associate with chemical graphs is called topological 

index. All graphs under our consideration are simple, connected and undirected. Let G = (V (G), E(G)) 

be a graph with order n = |V (G)| and size m = |E(G)|. The degree of a vertex vi in G is the number of 

edges incident on vi and is denoted by d(vi) or simply di. The corona 𝐺1⊙𝐺2 of two graphs G1 and G2 

is defined as the graph G obtained by taking one copy of G1(which has p1 points) and p1 copies of G2, 

and then joining the i th point of G1 to every point in the i th copy of G2. We follow [2] for more 

terminologies and notations not mentioned here. 

Definition 1. A molecular graph G = (V, E) is a simple graph having p = |V (G)| vertices and                      

q = |E(G)| edges. The vertices vi ∈ V represent non-hydrogen atoms and and the edges (vi, vj) ∈ E 

represent covalent bonds between the corresponding atoms. In particular, hydrocarbons are formed 

only by carbon and hydrogen atoms and their molecular graphs represent the carbon skeleton of the 

molecule. 

Definition 2. [3] Let G be a graph, then the first and second Zagreb indices of G are defined as  

𝑀1(𝐺) = ∑ 𝑑(𝑢)

𝑢∈𝑉(𝐺)

+ 𝑑(𝑣) 



Communications on Applied Nonlinear Analysis 

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𝑀2(𝐺) = ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸(𝐺)

 . 

Definition 3. [4] The Harmonic index of a graph G is defined as 

𝐻(𝐺) = ∑
2

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸(𝐺)

 . 

Definition 4. [5] The Geometric-Arithmetic index of a graph G is defined as   

   

𝐺𝐴(𝐺) = ∑
2√𝑑(𝑢)𝑑(𝑣)

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸(𝐺)

 

A significant field of graph theory is graph operations, which involves creating new graphs from 

existing ones by carrying out a series of actions on the basis of some graph theoretical parameters. 

They comprise unary and binary operations. There are several operations on two graphs G1 and G2 

which result in a graph G whose set of vertices is the cartesian product V1 × V2, where Vk is the vertex 

set of Gk. These include the cartesian product, the composition, tensor product etc.  

In 2024, [1] Akhil B., Roy John, Manju V.N. and G. Suresh Singh introduced a new graph product 

called ARMS-product of two graphs G1 and G2 as follows: 

Definition 5. Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs with order m and n respectively. The 

ARMS-Product of G1 and G2 denoted by G1 ⊠ G2 is a graph G with vertex set V = V1 ×V2 and two 

vertices x = (ui, vj) and y = (uk, vl) are adjacent in G if: 

1. either  𝑢𝑖 = 𝑢𝑘  𝑎𝑛𝑑  gcd (𝑑𝐺2(𝑣𝑗), 𝑑𝐺2(𝑣𝑙)) = 1 or 

2. gcd (𝑑𝐺1(𝑢𝑖), 𝑑𝐺1(𝑢𝑘)) = 1  and  𝑣𝑗 = 𝑣𝑙 

 𝑓𝑜𝑟 𝑎𝑙𝑙  𝑖, 𝑘 = 1,2, … ,𝑚  𝑤𝑖𝑡ℎ 𝑖 ≠ 𝑘 𝑎𝑛𝑑  𝑗, 𝑙 = 1,2, … , 𝑛  𝑤𝑖𝑡ℎ  𝑗 ≠ 𝑙. 

 Keeping the importance of this newly introduced graph product, in this paper various 

topological indices of graphs obtained through ARMS-product is evaluated. In [1] it is shown that the 

ARMS-product of two connected graph need not be connected. Therefore, we are considering only 

those graphs which produce connected graphs as an outcome and hence compute their topological 

indices in detail. 

2. Topological Indices of ARMS-Product of Graph 

In this section topological indices namely first, second Zagreb indices, Harmonic index and Geometric-

Arithmetic index of certain graphs obtained through the ARMS-Product of certain graphs are 

determined. 

Theorem 2.1. Let B be the bull graph and P2 be a path on two vertices. Then for the graph B ⊠ P2, 

first and second Zagreb indices, Harmonic index and Geometric-Arithmetic index are respectively 

given by: 

1.      𝑀1(𝐵 ⊠ 𝑃2) = 214. 



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2.     𝑀2(𝐵 ⊠ 𝑃2) = 497. 

3.     𝐻( 𝐵 ⊠ 𝑃 2) =
149

30
.  

4.    𝐺𝐴(𝐵 ⊠ 𝑃 2) = 11 +
16√5

3
. 

Proof: The ARMS-product of the bull graph, B and the path graph, P2 is a graph with 10 vertices. 

 

      B ⊠ P2 

The edge set of B ⊠ P2 can be partitioned as follows: 

 𝐸1 = {𝑢𝑣 | 𝑑𝑢 = 5, 𝑑𝑣 = 5}, |𝐸1| = 9. 

   𝐸2 = {𝑢𝑣 | 𝑑𝑢 = 5, 𝑑𝑣 = 4}, |𝐸2| = 12. 

𝐸3 = {𝑢𝑣 | 𝑑𝑢 = 4, 𝑑𝑣 = 4}, |𝐸3| = 2. 

 

                       𝑀1(𝐵 ⊠ 𝑃2)        =  ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸(𝐵⊠𝑃2)

+ 𝑑(𝑣) 

= ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸1

+ 𝑑(𝑣) + ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸2

+ 𝑑(𝑣) + ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸3

+ 𝑑(𝑣) 

= ∑ 5

𝑢𝑣∈𝐸1

+ 5 + ∑ 5

𝑢𝑣∈𝐸2

+ 4 + ∑ 4

𝑢𝑣∈𝐸3

+ 4 



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= 10 ∑ 1

𝑢𝑣∈𝐸1

+ 9 ∑ 1

𝑢𝑣∈𝐸2

+ 8 ∑ 1

𝑢𝑣∈𝐸3

 

= 10 × 9 + 9 × 12 + 8 × 2 

= 214. 

                           𝑀2(𝐵 ⊠ 𝑃2)    =  ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸(𝐵⊠𝑃2)

 

= ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸1

+ ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸2

+ ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸3

 

= ∑ 25

𝑢𝑣∈𝐸1

+ ∑ 20

𝑢𝑣∈𝐸2

+ ∑ 16

𝑢𝑣∈𝐸3

 

= 25 ∑ 1

𝑢𝑣∈𝐸1

+ 20 ∑ 1

𝑢𝑣∈𝐸2

+ 16 ∑ 1

𝑢𝑣∈𝐸3

 

= 25 × 9 + 20 × 12 + 16 × 2  

= 497. 

𝐻(𝐵 ⊠ 𝑃2)     = ∑
2

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸(𝐵⊠𝑃2)

 

= ∑
2

10
𝑢𝑣∈𝐸1

+ ∑
2

9
𝑢𝑣∈𝐸2

+ ∑
2

8
𝑢𝑣∈𝐸3

 

=
2

10
∑ 1

𝑢𝑣∈𝐸1

+
2

9
∑ 1

𝑢𝑣∈𝐸2

+
2

8
∑ 1

𝑢𝑣∈𝐸3

 

=
149

30
. 

             𝐺𝐴 (𝐵 ⊠ 𝑃2)   = ∑
2√𝑑(𝑢)𝑑(𝑣)

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸(𝐵⊠𝑃2)

 

= ∑
2√25

10
𝑢𝑣∈𝐸1

+ ∑
2√20

9
𝑢𝑣∈𝐸2

+ ∑
2√16

8
𝑢𝑣∈𝐸3

 

= ∑ 1

𝑢𝑣∈𝐸1

+
4√5

9
∑ 1

𝑢𝑣∈𝐸2

+ ∑ 1

𝑢𝑣∈𝐸3

 

= 9 + 12
4√5

9
+ 2 

= 11 +
16√5

3
. 



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This completes the proof. 

Theorem 2.2. Let K1,n and K1,m be two bipartite graphs. Then for the graph K1,n ⊠ K1,m first and 

second Zagreb indices, Harmonic index and Geometric-Arithmetic index are respectively given by: 

1. 𝑀1(𝐾1,𝑛⊠𝐾1,𝑚) = (𝑚 + 𝑛)2(𝑚 + 1)(𝑛 + 1). 

2.𝑀2(𝐾1,𝑛⊠𝐾1,𝑚) =
(𝑚 + 𝑛)2(𝑚 + 1)(𝑛 + 1)

2
. 

3. 𝐻(𝐾1,𝑛⊠𝐾1,𝑚) =
(𝑚 + 1)(𝑛 + 1)

2
. 

4. 𝐺𝐴(𝐾1,𝑛⊠𝐾1,𝑚) =
(𝑚 + 𝑛)(𝑚 + 1)(𝑛 + 1)

2
 

Proof: The ARMS- product of K1,n and K1,m is a (m + n) -regular graph on (m + 1)(n + 1) vertices. 

The number of edges of K1,n ⊠ K1,m is  
(𝑚+𝑛)(𝑚+1)(𝑛+1)

2
. 

𝑀1(𝐾1,𝑛⊠𝐾1,𝑚)        = ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

+ 𝑑(𝑣) 

= ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

+ 𝑑(𝑣) 

= ∑ 2(𝑚 + 𝑛)

𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

= 2(𝑚 + 𝑛)
(𝑚 + 𝑛)(𝑚 + 1)(𝑛 + 1)

2
 

= (𝑚 + 𝑛)2(𝑚 + 1)(𝑛 + 1). 

                                  𝑀2(𝐾1,𝑛⊠𝐾1,𝑚) = ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

= ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

= ∑ (𝑚 + 𝑛)2

𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

= (𝑚 + 𝑛)2 ×
(𝑚 + 𝑛)(𝑚 + 1)(𝑛 + 1)

2
 

=
(𝑚 + 𝑛)2(𝑚 + 1)(𝑛 + 1)

2
. 

 



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                                  𝐻(𝐾1,𝑛⊠𝐾1,𝑚)   = ∑
2

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

= ∑
2

2(𝑚 + 𝑛)
𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

=
1

𝑚 + 𝑛
∑ 1

𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

=
(𝑚 + 1)(𝑛 + 1)

2
. 

                            𝐺𝐴(𝐾1,𝑛⊠𝐾1,𝑚)        = ∑
2√𝑑(𝑢)𝑑(𝑣)

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

= ∑
2√(𝑚 + 𝑛)(𝑚 + 𝑛)

2(𝑚 + 𝑛)
𝑢𝑣∈𝐸(𝐾1,𝑛⊠𝐾1,𝑚)

 

=
(𝑚 + 𝑛)(𝑚 + 1)(𝑛 + 1)

2
. 

This completes the proof. 

Note: For 𝑛 =  3 and 𝑛 =  4, the ARMS- product of K1,3 and K1,4 is the rook graph and it is given 

below: 

 

             K1,3 ⊠ K1,4 

Theorem 2.3. Let Pn ⊙ K1 be a given graph and P2 be a path graph. Then for the graph                   

(Pn ⊙ K1) ⊠ P2,  first and second Zagreb indices, Harmonic index and Geometric-Arithmetic index 

are respectively given by: 

1.  𝑀1((𝑃𝑛⊙𝐾1) ⊠ 𝑃2) = {
100,                     𝑓𝑜𝑟 𝑛 = 2

𝑛(4𝑛 − 1)2, 𝑓𝑜𝑟  𝑛 > 3 .  
 



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2.  𝑀2((𝑃𝑛⊙𝐾1) ⊠ 𝑃2) = {
178,                                 𝑓𝑜𝑟 𝑛 = 2

𝑛2(3(2𝑛 − 1)2 + 4𝑛2), 𝑓𝑜𝑟  𝑛 > 3 .  
 

3.  𝐻((𝑃𝑛⊙𝐾1) ⊠ 𝑃2)  = {

83

21
,                                       𝑓𝑜𝑟 𝑛 = 2

 
2𝑛2

4𝑛 − 2
+
2𝑛(2𝑛 − 1)

4𝑛 − 1
+
𝑛

2
, 𝑓𝑜𝑟  𝑛 > 3.   

   

4.  𝐺𝐴((𝑃𝑛⊙𝐾1) ⊠ 𝑃2) =

{
 
 

 
 42 + (32√3)

7
,                𝑓𝑜𝑟 𝑛 = 2

2𝑛2 +
2𝑛(2𝑛 − 1)√(4𝑛2 − 2𝑛)

(4𝑛 − 1)
, 𝑓𝑜𝑟  𝑛 > 3.  

 

Proof: Case 1. 𝑛 =  2  

If 𝑛 =  2, consider (P2 ⊙ K1) ⊠ P2. 

 

(P2 ⊙ K1) ⊠ P2 

The edge set of (P2 ⊙ K1) ⊠ P2 can be partitioned as follows: 

 𝐸1 = {𝑢𝑣 | 𝑑𝑢 = 3, 𝑑𝑣 = 3}, |𝐸1| = 2. 

 𝐸2 = {𝑢𝑣 | 𝑑𝑢 = 3, 𝑑𝑣 = 4}, |𝐸2| = 8. 

𝐸3 = {𝑢𝑣 | 𝑑𝑢 = 4, 𝑑𝑣 = 4}, |𝐸3| = 4. 

𝑀1((𝑃2⊙𝐾1) ⊠ 𝑃2) = ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2)

+ 𝑑(𝑣) 

= ∑ 3

𝑢𝑣∈𝐸1

+ 3 + ∑ 3

𝑢𝑣∈𝐸2

+ 4 + ∑ 4

𝑢𝑣∈𝐸3

+ 4 

= 6 ∑ 1

𝑢𝑣∈𝐸1

+ 7 ∑ 1

𝑢𝑣∈𝐸2

+ 8 ∑ 1

𝑢𝑣∈𝐸3

 



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= 6 × 2 + 7 × 8 + 8 × 4 

= 100. 

𝑀2((𝑃2⊙𝐾1) ⊠ 𝑃2) = ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2)

𝑑(𝑣) 

= ∑ 9

𝑢𝑣∈𝐸1

+ ∑ 12

𝑢𝑣∈𝐸2

+ ∑ 16

𝑢𝑣∈𝐸3

 

= 9 × 2 + 12 × 8 + 16 × 4  

= 178. 

𝐻((𝑃2⊙𝐾1) ⊠ 𝑃2)   = ∑
2

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2)

 

= ∑
2

6
𝑢𝑣∈𝐸1

+ ∑
2

7
𝑢𝑣∈𝐸2

+ ∑
2

8
𝑢𝑣∈𝐸3

 

=
2

6
× 2 +

2

7
× 8 +

2

8
× 4 

=
83

21
. 

𝐺𝐴((𝑃2⊙𝐾1) ⊠ 𝑃2) = ∑
2 √𝑑(𝑢)𝑑(𝑣)

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸((𝑃2⊙𝐾1)⊠𝑃2)

 

= ∑
2√3 ×  3

6
𝑢𝑣∈𝐸1

+ ∑
2√3 ×  4

7
𝑢𝑣∈𝐸2

+ ∑
2√4 ×  4

8
𝑢𝑣∈𝐸3

 

= ∑ 1

𝑢𝑣∈𝐸1

+
4√3

7
∑ 1

𝑢𝑣∈𝐸2

+ ∑ 1

𝑢𝑣∈𝐸3

 

= 2 +
4√3

7
× 8 + 4 

=
42 + 32√3

7
 . 

Case 2. 𝑛 >  3  

If 𝑛 >  3, the edge set of (Pn ⊙ K1) ⊠ P2 can be partitioned as follows: 

 𝐸1 = {𝑢𝑣 | 𝑑𝑢 = 2𝑛 − 1, 𝑑𝑣 = 2𝑛 − 1}, |𝐸1| = 𝑛2. 

 𝐸2  = {𝑢𝑣 | 𝑑𝑢 = 2𝑛 − 1, 𝑑𝑣 = 2𝑛}, |𝐸2| = 2𝑛2 − 𝑛. 

𝐸3  = {𝑢𝑣 | 𝑑𝑢 = 2𝑛, 𝑑𝑣 = 2𝑛}, |𝐸3| = 𝑛
2. 



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𝑀1((𝑃𝑛⊙𝐾1) ⊠ 𝑃2) = ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2)

+ 𝑑(𝑣) 

= ∑ (2𝑛 − 1

𝑢𝑣∈𝐸1

) + (2𝑛 − 1) + ∑ (2𝑛 − 1)

𝑢𝑣∈𝐸2

+ (2𝑛) + ∑ (2𝑛)

𝑢𝑣∈𝐸3

+ (2𝑛) 

= (4𝑛 − 2) ∑ 1

𝑢𝑣∈𝐸1

+ (4𝑛 − 1) ∑ 1

𝑢𝑣∈𝐸2

+ (4𝑛) ∑ 1

𝑢𝑣∈𝐸3

 

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

= 𝑛(4𝑛 − 1)2. 

𝑀2((𝑃𝑛⊙𝐾1) ⊠ 𝑃2) = ∑ 𝑑(𝑢)

𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2)

𝑑(𝑣) 

= ∑ (2𝑛 − 1

𝑢𝑣∈𝐸1

)(2𝑛 − 1) + ∑ (2𝑛 − 1)

𝑢𝑣∈𝐸2

(2𝑛) + ∑ (2𝑛)

𝑢𝑣∈𝐸3

(2𝑛) 

= (2𝑛 − 1)2 ∑ 1

𝑢𝑣∈𝐸1

+ (4𝑛2 − 2𝑛) ∑ 1

𝑢𝑣∈𝐸2

+ (4𝑛2) ∑ 1

𝑢𝑣∈𝐸3

 

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

= 𝑛2(3(2𝑛 − 1)2 +  4𝑛2). 

𝐻((𝑃𝑛⊙𝐾1) ⊠ 𝑃2)   = ∑
2

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2)

 

= ∑
2

(4𝑛 − 2)
𝑢𝑣∈𝐸1

+ ∑
2

(4𝑛 − 1)
𝑢𝑣∈𝐸2

+ ∑
2

(4𝑛)
𝑢𝑣∈𝐸3

 

=
2

(4𝑛 − 2)
× 𝑛2 +

2

(4𝑛 − 1)
× (2𝑛2 − 𝑛) +

2

(4𝑛)
× 𝑛2 

=
𝑛2

(2𝑛 − 1)
+
2𝑛(2𝑛 − 1)

(4𝑛 − 1)
+
𝑛

2
. 

𝐺𝐴((𝑃𝑛⊙𝐾1) ⊠ 𝑃2) = ∑
2 √𝑑(𝑢) 𝑑(𝑣)

𝑑(𝑢) + 𝑑(𝑣)
𝑢𝑣∈𝐸((𝑃𝑛⊙𝐾1)⊠𝑃2)

 

= ∑
2√2𝑛 − 1 × 2𝑛 − 1

4𝑛 − 2
𝑢𝑣∈𝐸1

+ ∑
2√2𝑛 × 2𝑛 − 1

4𝑛 − 1
𝑢𝑣∈𝐸2

+ ∑
2√2𝑛 × 2𝑛

4𝑛
𝑢𝑣∈𝐸3

 

= ∑
2(2𝑛 − 1)

(4𝑛 − 2)
𝑢𝑣∈𝐸1

+ ∑
2√(4𝑛2 − 2𝑛)

(4𝑛 − 1)
𝑢𝑣∈𝐸2

+ ∑
2× 2𝑛

4𝑛
𝑢𝑣∈𝐸3

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2 (2025) 

 

263 
https://internationalpubls.com 

= 2𝑛2 +
2𝑛(2𝑛 − 1)√(4𝑛2 − 2𝑛)

(4𝑛 − 1)
. 

This completes the proof. 

Note: For 𝑛 = 3, first and second Zagreb indices, Harmonic index and Geometric-Arithmetic index 

of (𝑃3⊙𝐾1) ⊠ 𝑃2 are 388, 1106, 
986

165
  and 

198 +32√30

11
  respectively. 

3.  Conclusion 

In this work, we determined various topological indices of ARMS-product of certain classes of graphs. 

Since the ARMS-product is a newly introduced graph product, it has great significance in the field of 

topological indices. 

Acknowledgement  

The authors would like to thank University of Kerala for providing all facilities. The first author would 

like to thank the technical help received from DST-FIST facility of Department of Mathematics,          

St. Stephen’s College, Pathanapuram, Kollam, Kerala. The second author would like to thank The 

Kerala State Council for Science, Technology and Environment (KSCSTE) for the financial support. 

Further we would like to inform that all the authors contributed equally in this work. The authors would 

like to thank Dr. G. Suresh Singh, Professor, Department of Mathematics, University of Kerala for his 

continuous support and valuable suggestions. 

Declarations 

• Funding: This work was supported by Kerala State Council for Science, Technology and 

Environment (KSCSTE).  

• Authors’ contributions: All authors contributed the study conception and design. All authors read 

and approved the final manuscript. 

Conflicts of Interest: The authors declare that they have no conflicts of interest regarding the 

publication of this article. 

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