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Computing Topological Indices of Certain Networks 

 

Vidya S Umadi1, A M Sangogi2 
1Assistant Professor 

Department of Mathematics, Vidya  Samvardhak Mandal’s 

Somashekhar R Kothiwale Institute of Technology, Nipani, Karnataka State, India  vidyaumadi@gmail.com 
2Professor 

Department of Mathematics, SGBIT, Belagavi, Karnataka State, India. 

amsangogi@gmail.com 

 

Article History:  

Received: 01-03-2024 

Revised: 29-04-2024 

Accepted: 15-05-2024 

Abstract: 

In mathematical chemistry, topological indices are molecular descriptors that 

are calculated on the molecular graph of a chemical compound. The 

molecular graph is a graph which is obtained from some chemical structures. 

The degree of every molecular graph cannot exceeds 4. Topological indices 

are numerical quantities of a graph that describe its topology. An atom 

represents a vertex and a bond between two atoms represents an edge in a 

molecular graph. Mainly there are three types of topological indices viz., 

degree-based, distance based and eigenvalue-based topological indices. The 

first degree-based topological indices are the first and second Zagreb indices. 

The first Zagreb index M_1 is defined as the sum of squares of degrees of 

each vertex in a graph G and the second Zagreb index M_2 is the product of 

degree of every adjacent vertices. In this case the summation goes on the set 

of edges of a graph G. The most studied topological indices are degree-based 

topological indices. Motivated by these topological indices in this paper, we 

introduce five new degree-based topological indices based on the 

neighborhood degree of a vertex. Further, we compute the values of various 

nanostructures like hexagonal parallelogram P(m,n) nanotube, triangular 

benzenoid G_n,zigzag-edge coronoid fused with starphene nanotubes 

ZCS(k,l,m), dominating derived networks D_1,D_2,D_3, Porphyrin 

Dendrimer, Zinc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimer, 

Poly(Ethylene amido amine Dendrimer, PAMAM 

dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1), linear polyomino chain L_n,Z_n,B_n^1 

(n≥3),B_n^2 (n≥3) and triangular, hourglass, and jagged-rectangle benzenoid 

systems of these indices. The standard computational techniques are used for 

the computation of topological indices of nanostructures. For the edge 

partition of the nanostructures the algebraic techniques are used. Using these 

techniques computation of topological indices became easy and also helped 

to get the more accurate results. 

Keywords: Molecular Graph, Nanostructures, Dendrimers, Topological 

indices. 

 

1. Introduction 

In the realm of modern chemistry, the quest to understand the intricate structures of 

molecules and their impact on chemical properties has led researchers to explore various 

analytical tools and methodologies. Among these, the field of mathematical chemistry stands 

out for its emphasis on applying mathematical concepts and techniques to unravel the 



Communications on Applied Nonlinear Analysis 

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mysteries of molecular structures. Central to this endeavor is the study of molecular graphs 

and their characterization through topological indices, which serve as powerful descriptors 

of molecular topology. Topological indices represent numerical quantities derived from the 

molecular graph of a chemical compound. The molecular graph itself is a representation of 

the chemical structure, wherein atoms are depicted as vertices and bonds as edges. By 

analyzing the connectivity patterns and geometrical arrangements within these graphs, 

researchers can gain valuable insights into the structural features that influence the behavior 

of molecules in various chemical contexts[1]. 

Within the framework of topological indices, three main categories emerge: degree-based, 

distance-based, and eigenvalue-based indices. Each category offers unique perspectives on 

molecular topology, with degree-based indices being particularly prominent due to their 

simplicity and effectiveness in capturing essential structural information. One of the 

cornerstone degree-based indices is the Zagreb indices, comprising the first and second 

Zagreb indices. The first Zagreb index, denoted as M1M1[2], quantifies the sum of the 

squares of vertex degrees in the molecular graph, providing a measure of overall 

connectivity. On the other hand, the second Zagreb index, denoted as M2M2, captures the 

product of degrees of adjacent vertices, thus highlighting local structural motifs within the 

molecule[3]. 

Building upon the foundational concepts of degree-based indices, this paper introduces five 

novel topological indices rooted in the notion of neighborhood degree. The neighborhood 

degree of a vertex reflects the cumulative degree of its neighboring vertices, offering 

insights into the local structural environment of each vertex. By incorporating this concept 

into the design of new indices, the paper aims to enrich the repertoire of tools available for 

analyzing molecular graphs and uncovering subtle structural variations[4]. Beyond 

theoretical development, this paper also emphasizes the practical applications of topological 

indices to a diverse array of nanostructures. Nanostructures, characterized by their unique 

geometries and properties at the nanoscale, present intriguing challenges and opportunities 

for topological analysis. Examples of such nanostructures include hexagonal parallelogram 

nanotubes, benzenoid systems, dendrimers, and polyomino chains, among others. By 

computing the proposed indices for these nanostructures, the paper seeks to demonstrate 

their effectiveness in capturing the complex topology inherent in these systems.To facilitate 

the computation of topological indices for nanostructures, the paper employs a combination 

of standard computational techniques and algebraic methods for edge partitioning. These 

techniques not only streamline the calculation process but also enhance the accuracy and 

reliability of the results obtained. By leveraging computational and algebraic tools, 

researchers can explore the intricate details of molecular topology with greater efficiency 

and precision[5]. 

    Introduction of Novel Degree-Based Topological Indices: The primary objective of this 

paper is to introduce five new degree-based topological indices that are based on the concept 

of neighborhood degree. These indices are designed to provide a more nuanced 

characterization of molecular graphs, with a focus on capturing local structural information. 



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Application to Various Nanostructures: Another objective is to demonstrate the applicability 

of the proposed indices to a diverse range of nanostructures, including nanotubes, benzenoid 

systems, dendrimers, and polyomino chains. By computing these indices for different 

nanostructures, the paper aims to showcase their utility in analyzing complex molecular 

architectures[6,7].  Utilization of Computational and Algebraic Techniques: The paper aims 

to leverage standard computational techniques for the computation of topological indices, 

supplemented by algebraic methods for edge partitioning in nanostructures. This approach is 

expected to enhance the efficiency and accuracy of the calculations, facilitating more robust 

analysis and interpretation of the results[8]. 

Overall, the objectives of the paper encompass both theoretical advancements in topological 

index theory and practical applications to real-world nanostructures, with a focus on 

enhancing our understanding of molecular topology in mathematical chemistry[9,10].In this 

section, we consider chemical structures like hexagonal parallelogram 𝑃(𝑚, 𝑛) nanotube, 

triangular benzenoid 𝐺𝑛,zigzag-edge coronoid fused with starphene nanotubes 𝑍𝐶𝑆(𝑘, 𝑙, 𝑚), 

dominating derived networks 𝐷1, 𝐷2, 𝐷3, Porphyrin Dendrimer, Zinc-Porphyrin Dendrimer, 

Propyl Ether Imine Dendrimer, Poly(Ethylene amido amine Dendrimer, PAMAM 

dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1), linear polyomino chain 𝐿𝑛, 𝑍𝑛, 𝐵𝑛
1(𝑛 ≥ 3), 𝐵𝑛

2(𝑛 ≥ 3)  and 

triangular, hourglass, and jagged-rectangle benzenoid systems [2,3,4,5]. For notations and 

terminology used in this paper are taken from. 

2. Related Works 

Topological indices have got importance due to its applications in chemistry as well as in 

life science. Recently, Hosamani et al. [11], have put forward the following new degree-

based topological indices: 

 𝑆1(𝐺) = ∑ 𝑑(𝑣𝑖)
|𝑑(𝑣𝑖)|

𝑣𝑖∈𝑉                        (1) 

     𝑆2(𝐺) = ∑ (𝑑(𝑣𝑖)|𝑑(𝑣𝑖)|
𝑣𝑖𝑣𝑗∈𝐸 + 𝑑(𝑣𝑗)|𝑑(𝑣𝑗)|)                  (2) 

     𝑆3(𝐺) = ∑ (𝑑(𝑣𝑖)|𝑑(𝑣𝑗)|
𝑣𝑖𝑣𝑗∈𝐸 + 𝑑(𝑣𝑗)|𝑑(𝑣𝑖)|)                  (3) 

     𝑆4(𝐺) = ∑ (𝑑(𝑣𝑖)|𝑑(𝑣𝑖)|
𝑣𝑖𝑣𝑗∈𝐸 + 𝑑(𝑣𝑗)|𝑑(𝑣𝑗)|)                  (4) 

    𝑆5(𝐺) = ∑ (𝑑(𝑣𝑖)|𝑑(𝑣𝑗)|
𝑣𝑖𝑣𝑗∈𝐸 + 𝑑(𝑣𝑗)|𝑑(𝑣𝑖)|)                        (5) 

3. Methodology 

Motivated by the inverse degree of a vertex and the above-mentioned topological 

indices, here we introduced the following topological indices in the field of chemical graph 

theory. 

𝑅𝑆1(𝐺) = ∑
1

𝑑(𝑣𝑖)|𝑑(𝑣𝑖)|𝑣𝑖∈𝑉                                                                      (6) 

𝑅𝑆2(𝐺) = ∑
1

𝑑(𝑢)|𝑑(𝑢)|+𝑑(𝑣)|𝑑(𝑣)|𝑢𝑣∈𝐸                                                         (7) 

𝑅𝑆3(𝐺) = ∑
1

𝑑(𝑢)|𝑑(𝑣)|+𝑑(𝑣)|𝑑(𝑢)|𝑢𝑣∈𝐸                                                         (8) 

𝑅𝑆4(𝐺) = ∑
1

𝑑(𝑢)|𝑑(𝑢)|𝑑(𝑣)|𝑑(𝑣)|𝑢𝑣∈𝐸                                                            (9) 



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𝑅𝑆5(𝐺) = ∑
1

𝑑(𝑢)|𝑑(𝑣)|𝑑(𝑣)|𝑑(𝑢)|𝑢𝑣∈𝐸                                                          (10) 

 

In this paper, we consider their chemical structures like hexagonal parallelogram 𝑃(𝑚, 𝑛) 

nanotube, triangular benzenoid 𝐺𝑛, zigzag-edge coronoid fused with starphene nanotubes 

𝑍𝐶𝑆(𝑘, 𝑙, 𝑚), dominating derived networks 𝐷1, 𝐷2, 𝐷3, Porphyrin Dendrimer, Zinc-Porphyrin 

Dendrimer, Propyl Ether Imine Dendrimer, Poly(Ethylene amido amine Dendrimer, PAMAM 

dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1), linear polyomino chain 𝐿𝑛, 𝑍𝑛, 𝐵𝑛
1(𝑛 ≥ 3), 𝐵𝑛

2(𝑛 ≥ 3)  and 

triangular, hourglass, and jagged-rectangle benzenoid systems which are depicted in the 

following figures 2.1, 2.2, 2.3, 2.4, 2.5 and  2.6 respectively: 

 

Figure 2.1: hexagonal parallelogram . 

 

Figure 2.2: Triangular benzenoid . 



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Figure 2.3: The zigzag-edge coronoid fused with starphene.  

 

Figure 2.4: Dominating derived network. 

 

Figure 2.5: Porphyrin dendrimer. 



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Figure 2.6: Benzoid hourglass system. 

4. Results and Discussion 

Theorem 1. Let 𝐺 denotes the line graph of subdivision graph of the hexagonal 

parallelogram, then 

 

𝑅𝑆2(𝐺) =
1

2
𝑚𝑛 +

1145

3348
(𝑚 + 𝑛) +

1087

1674
    (11) 

𝑅𝑆3(𝐺) =
1

6
𝑚𝑛 +

823

1836
(𝑚 + 𝑛) +

401

918
    (12) 

𝑅𝑆4(𝐺) =
1

81
𝑚𝑛 +

929

5832
(𝑚 + 𝑛) +

611

1458
   (13) 

𝑅𝑆5(𝐺) =
1

81
𝑚𝑛 +

1037

5832
(𝑚 + 𝑛) +

557

1458
   (14) 

 

Proof. Let (𝑚,);,∈ℤ+ be a hexagonal parallelogram of order 2(𝑚+ 𝑛+ 𝑚𝑛i and size (3𝑚𝑛+ 

2𝑚+ 2𝑛+ 1) respectively. Let 𝐺= 𝐿(𝑆(𝑃(𝑚,𝑛))) denotes the line graph of subdivision graph 

of 𝑃(𝑚,𝑛);𝑚,𝑛∈ℤ+ then clearly, the order and size of 𝐺are 2(3mn+2m+2n+1) and  

9𝑚𝑛+ 4𝑚+ 4𝑛+ 5 respectively. The edge set of 𝐺 can be partitioned into three disjoint sets 

𝜀2,2,𝜀2,3 andi 𝜀3,3, where 𝜀(𝐿(𝑆(𝑃(𝑚,𝑛)))) =𝜀2,2∪𝜀2,3∪𝜀3,3. Further, ∣𝜀2,2∣ = 2(𝑚+𝑛+4), ∣𝜀2,3∣ = 

4(𝑚+𝑛−2), ∣𝜀3,3∣ = 9𝑚𝑛−2𝑚−2𝑛−5. Such that ∣𝜀(𝐿(𝑆(𝑃(𝑚,𝑛))))∣ = ∣𝜀2,2∣+∣𝜀2,3∣+∣𝜀3,3∣ = 

9𝑚𝑛+4𝑚+4𝑛+5. Thus, with this background by employing equations (1)-(5) and (6)-(10), 

we get the required results.  

Theorem 2. Let ((𝐺𝑛)) denotes their line graphic of subdivision graph of the hexagonal 

parallelogram, then 

𝑅𝑆2(𝐿(𝑆(𝐺𝑛))) =
83

1000
𝑛2 +

3

5
𝑛 +

41

50
    (15) 

𝑅𝑆3(𝐿(𝑆(𝐺𝑛))) =
9

108
𝑛2 +

19

25
𝑛 −

9

100
    (16) 

𝑅𝑆4(𝐿(𝑆(𝐺𝑛))) =
9

1458
𝑛2 +

1

4
𝑛 +

3

25
    (17) 

𝑅𝑆5(𝐿(𝑆(𝐺𝑛))) =
9

1458
𝑛2 +

27

100
𝑛 +

47

100
   (18) 



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Proof. Let 𝐺𝑛;∈ℤ+ be a triangular benzenoid of order 𝑛2+ 4𝑛+ 1 and size 
3

2
𝑛(𝑛 +

3)respectively. Let (𝑆(𝐺𝑛))) denotes the line graph of subdivision graph of 𝐺𝑛  then clearly, 

the order and size of 𝐿(𝑆(𝐺𝑛))) are 3𝑛(𝑛+ 3) and 
3(3𝑛2+7𝑛+2)

2
  respectively. The edge set of 

(𝑆(𝐺𝑛))) can be partitioned into three disjoint sets 𝜀2,2,𝜀2,3 and 𝜀3,3, where 

(𝐿(𝑆(𝐺𝑛)))) = 𝜀2,2 ∪𝜀2,3 ∪𝜀3,3. Further, ∣𝜀2,2∣ = 3(𝑛+ 3), ∣𝜀2,3∣ = 6(𝑛− 1), ∣𝜀3,3∣ =
3(3𝑛2+𝑛−4)

2
. 

Such that 

|𝜀 (𝐿(𝑆(𝐺𝑛)))| = |𝜀2,2| + |𝜀2,3| + |𝜀3,3| =
3(3𝑛2 + 7𝑛 + 2)

2
. 

 Thus, with thisbackground by employing equations (1)-(5) and (6)-(10), we get the required 

results.  

Theorem 3. Let ((𝐼)) be the line graph of the subdivision graph of zigzag-edges coronoid 

fused with starphene nanotubes 𝑍𝐶𝑆(𝑘,𝑙,𝑚) for 𝑘= 𝑙= 𝑚= 4. Then 

𝑅𝑆2(𝐿(𝑆(𝐼))) =
3

2
(𝑘 + 𝑙 + 𝑚) −

355

1116
    (19) 

𝑅𝑆3(𝐿(𝑆(𝐼))) =
1129

612
(𝑘 + 𝑙 + 𝑚) −

941

100
   (20) 

𝑅𝑆4(𝐿(𝑆(𝐼))) =
103

200
(𝑘 + 𝑙 + 𝑚) −

271

100
    (21) 

𝑅𝑆5(𝐿(𝑆(𝐼))) =
57

100
(𝑘 + 𝑙 + 𝑚) −

31

10
    (22) 

 

Proof. Let be zizag-edge coronoid fused with starphene nanotubes (𝑘,,) for 𝑘= 𝑙= 𝑚= 4 of 

order 36𝑘+ 54 and size 15(𝑘+ 𝑙+ 𝑚) − 63 respectively. Let ((𝐼)) be the line graphic of the 

subdivision graph of zigzag-edge coronoid fused with starphene nanotubes (𝑘,,𝑚) for 𝑘= 𝑙= 

𝑚= 4. Theni clearly, the order and size of ((𝐼)) are 30(𝑘+𝑙+𝑚126) and 39(𝑘+𝑙+𝑚)+153 

respectively. The edge set of 𝐿(𝑆(𝐼)) can be partitioned into three disjoint sets 𝜀2,2,𝜀2,3 and 

𝜀3,3,where 𝜀(𝐿(𝑆(𝐼))) = 𝜀2,2i ∪𝜀2,3 ∪𝜀3,3. Further, ∣𝜀2,2∣ = 6(𝑘+ 𝑙+𝑚− 5), ∣𝜀2,3∣ = 12(𝑘+ 𝑙+ 𝑚− 

7), ∣𝜀3,3∣ = 21(𝑘+ 𝑙+ 𝑚) − 39. Such that ∣(𝐿(𝑆(𝐼)))∣ = ∣𝜀2,2∣ + ∣𝜀2,3∣ + ∣𝜀3,3∣ = 39(𝑘+ 𝑙+ 𝑚) + 

153. Thus, with this background by employing equations (1)-(5) and (6)-(10),we get the 

required results.  

Theorem 4.  Let 𝐷1(𝑛) be the dominating derived network of 1st type. Then 

 

𝑅𝑆2(𝐷1(𝑛)) =
9

25
𝑛2 +

1

5
𝑛 +

3

100
    (23) 

𝑅𝑆3(𝐷1(𝑛)) =
49

100
𝑛2 +

19

25
𝑛 −

8

25
    (24) 

𝑅𝑆4(𝐷1(𝑛)) =
9

500
𝑛2 +

29

100
𝑛 −

21

500
    (25) 

𝑅𝑆5(𝐷1(𝑛)) =
1

50
𝑛2 +

39

10
𝑛 −

11

100
    (26) 

 

Proof. Let 𝐷1(𝑛) be the dominating derived network of 1st type. The edge set of 𝐷1(𝑛) can 

be partitioned into six disjoint sets 𝜀2,2,𝜀2,3,𝜀2,4,𝜀3,3,𝜀3,4 and 𝜀4,4, where 𝜀(𝐷1(𝑛)) = 



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𝜀2,2∪𝜀2,3 ∪𝜀2,4 ∪𝜀3,3 ∪𝜀3,4 ∪𝜀4,4. Further, ∣𝜀2,2∣ = 4𝑛, ∣𝜀2,3∣ = 4𝑛− 4, ∣𝜀2,4∣ = 28𝑛 16 ∣𝜀3,3∣ = 9𝑛2 − 

13𝑛+ 5, ∣𝜀3,4∣ = 36𝑛2 − 56𝑛+ 24 and ∣𝜀4,4∣ = 36𝑛2− 52𝑛+20. Such that 

∣𝜀(𝐷1(𝑛))∣ = ∣𝜀2,2∣∣𝜀2,3∣ + ∣𝜀2,4∣ + ∣𝜀3,3∣ +i ∣𝜀3,4∣+ ∣𝜀4,4∣. Thus, with this background by 

employing equations (1)-(5) and (6)-(10), we get the required results.  

Theorem 5. Let 𝐷2(𝑛) be the dominating derived network of 2nd type. Then 

𝑅𝑆2(𝐷2(𝑛)) =
79

100
𝑛2 −

2

5
𝑛 +

13

50
    (27) 

𝑅𝑆3(𝐷2(𝑛)) =
69

50
𝑛2 −

41

100
𝑛 +

3

50
    (28) 

𝑅𝑆4(𝐷2(𝑛)) =
17

100
𝑛2 +

13

200
𝑛 +

11

250
    (29) 

𝑅𝑆5(𝐷2(𝑛)) =
13

50
𝑛2 +

21

500
𝑛 +

13

500
    (30) 

 

Proof. Let 𝐷2(𝑛) be the dominating derived network of 2nd type. The edge set of 𝐷1(𝑛) can 

be partitioned into five disjoint sets 𝜀2,2,𝜀2,3,𝜀2,4,𝜀3,4 and 𝜀4,4, where  

(𝐷2(𝑛)) =𝜀2,2 ∪𝜀2,3 ∪𝜀2,4 ∪𝜀3,4 ∪𝜀4,4.  Further, ∣𝜀2,2∣ = 4𝑛, ∣𝜀2,3∣ = 18𝑛2 −22𝑛+6,i ∣𝜀2,4∣= 

28𝑛−16, 

∣𝜀3,4∣ = 36𝑛2 − 56𝑛+ 24 and ∣𝜀4,4∣ = 36𝑛2 −52𝑛+ 20. Such that ∣(𝐷2(𝑛))∣ = ∣𝜀2,2∣ + ∣𝜀2,3∣ + ∣𝜀2,4∣ 

+ ∣𝜀3,4∣ + ∣𝜀4,4∣.  

Thus, with this background by employing equations(1)-(5) and (6)-(10), we get the required 

results.  

Theorem 6.  Let 𝐷3(𝑛) be the dominating derived network of 3rd type. Then 

𝑅𝑆2(𝐷3(𝑛)) =
7

25
𝑛2 +

21

100
𝑛 +

17

200
    (31) 

𝑅𝑆3(𝐷3(𝑛)) =
127

100
𝑛2 −

17

50
𝑛 +

43

500
    (32) 

𝑅𝑆4(𝐷3(𝑛)) =
9

250
𝑛2 +

23

100
𝑛 +

7

1000
    (33) 

𝑅𝑆5(𝐷3(𝑛)) =
141

1000
𝑛2 +

17

100
𝑛 +

7

1000
    (34) 

 

Proof. Let 𝐷3(𝑛 ) be the dominating derived network of 3rd type. The edge set of 𝐷1(𝑛) can 

be partitioned into three disjoint sets 𝜀2,2,𝜀2,4 and 𝜀4,4, where 𝜀(𝐷3(𝑛)) = 𝜀2,2 ∪𝜀2,4 ∪ 

𝜀4,4. Further, ∣𝜀2,2∣ = 4𝑛, ∣𝜀2,4∣ = 36𝑛2 − 20𝑛 and ∣𝜀4,4∣ = 72𝑛2 − 108𝑛+ 44. Such that ∣(𝐷3(𝑛))∣ 

= ∣𝜀2,2∣ + ∣𝜀2,4∣ + ∣𝜀4,4∣. Thus, with this background by employing equations 

(1)-(5) and (6)-(10), we get the required results.  

Theorem 7. Let 𝐷𝑛𝑃𝑛 be the prophyrin dendrimer. Then 

𝑅𝑆2(𝐷𝑛𝑃𝑛) =
323

100
𝑛 −

41

50
    (35) 

𝑅𝑆3(𝐷𝑛𝑃𝑛) =
967

100
𝑛 −

49

50
    (36) 

𝑅𝑆4(𝐷𝑛𝑃𝑛) =
63

50
𝑛 −

37

100
    (37) 



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𝑅𝑆5(𝐷𝑛𝑃𝑛) =
399

50
𝑛 −

79

200
    (38) 

Proof. Let 𝐷𝑛𝑃𝑛be the prophyrin dendrimer of order 96𝑛−10 and size 105𝑛−11 respectively. 

The edge set of 𝐷𝑛𝑃𝑛 can be partitioned into six disjoint sets 𝜀1,3,𝜀1,4,𝜀2,2,𝜀2,3,𝜀3,3 and 𝜀3,4, 

where 𝜀(𝐷𝑛𝑃𝑛) = 𝜀1,3 ∪𝜀1,4 ∪𝜀2,2 ∪𝜀2,3 ∪𝜀3,3∪𝜀3,4. Further, ∣𝜀1,3∣= 2𝑛, ∣𝜀1,4∣ = 24𝑛, ∣𝜀2,2∣= 

10𝑛− 5, ∣𝜀2,3∣ = 48𝑛− 6, ∣𝜀3,3∣ = 13𝑛 and ∣𝜀3,4∣ = 8𝑛. Such that ∣(𝐷𝑛𝑃𝑛)∣ = ∣𝜀1,3∣ + ∣𝜀1,4∣ +∣𝜀2,2∣ 

∣𝜀2,3∣ + ∣𝜀3,3∣ + ∣𝜀3,4∣ = 105𝑛− 11. Thus, with this background by employing equations (1)-(5) 

and (6)-(10), we get the required results.  

Theorem 8. Let 𝐷𝑃𝑍𝑛 be the Zinc-Porphyrin dendrimer. Then 

𝑅𝑆2(𝐷𝑃𝑍𝑛) =
343

100
2𝑛 −

13

10
    (39) 

𝑅𝑆3(𝐷𝑃𝑍𝑛) =
9

2
2𝑛 −

171

100
    (40) 

𝑅𝑆4(𝐷𝑃𝑍𝑛) =
69

50
2𝑛 −

21

50
    (41) 

𝑅𝑆5(𝐷𝑃𝑍𝑛) =
8

5
2𝑛 −

1

2
     (42) 

Proof. Let 𝐷𝑃𝑍𝑛 be the Zinc-Porphyrin dendrimer. of order 96𝑛−10 and size 105𝑛−11 

respectively. The edge set of 𝐷𝑃𝑍𝑛can be partitioned into four disjoint sets 𝜀2,2,𝜀2,3,𝜀3,3 and 

𝜀3,4, where 𝜀(𝐷𝑃𝑍𝑛) = 𝜀2,2 ∪𝜀2,3 ∪𝜀3,3 ∪𝜀3,4. Further, ∣𝜀2,2∣ = 16 ⋅ 2𝑛− 4, ∣𝜀2,3∣ = 40 ⋅ 2𝑛− 16, 

∣𝜀3,3∣ = 8 ⋅ 2𝑛− 16i and ∣𝜀3,4∣ = 4. Such that ∣(𝐷𝑃𝑍𝑛)∣ = ∣𝜀2,2∣ + ∣𝜀2,3∣ + ∣𝜀3,3∣ + ∣𝜀3,4∣ = 105𝑛− 

11.Thus,with this background by employing equations (1)-(5) and (6)-(10),we get the 

required results.  

Theorem 9. For the PAMAM dendrimers 𝑃𝐷1, we have 

𝑅𝑆2(𝑃𝐷1) =
37

10
2𝑛 −

81

50
     (43) 

𝑅𝑆3(𝑃𝐷1) =
599

100
2𝑛 −

13

5
    (44) 

𝑅𝑆4(𝑃𝐷1) =
23

10
2𝑛 −

39

50
     (45) 

𝑅𝑆5(𝑃𝐷1) =
59

12
2𝑛 −

173

100
    (46) 

Proof. Let 𝑃𝐷1 denote PAMAM dendrimers with trifunctional core unit generated by 𝐺𝑛 with 

𝑛 growth stages. The edge set of 𝑃𝐷1 can be partitioned into four disjoint sets 

𝜀1,2,𝜀1,3,𝜀2,2 and 𝜀2,3, where 𝜀(𝑃𝐷1) = 𝜀1,2∪𝜀1,3 ∪𝜀2,2 ∪𝜀2,3. Further, ∣𝜀1,2∣ = 3 ⋅ 2𝑛,∣𝜀,3∣ = 6 ⋅ 2𝑛− 

3, ∣𝜀2,2∣ = 18 ⋅ 2𝑛− 9 and ∣𝜀2,3∣ = 21 ⋅ 2𝑛− 12. Such that ∣(𝑃𝐷1)∣ = ∣𝜀1,2∣ + ∣𝜀1,3∣ + ∣𝜀2,2∣ + ∣𝜀2,3. 

Thus, with this background by employing equations (1)-(5) and (6)-(10), we get the required 

results.  

Theorem 10. For their PAMAM dendrimers 𝑃𝐷2, we have 

𝑅𝑆2(𝑃𝐷2) =
499

100
2𝑛 −

49

25
    (47) 

𝑅𝑆3(𝑃𝐷2) =
399

50
2𝑛 −

16

5
    (48) 

𝑅𝑆4(𝑃𝐷2) =
61

20
2𝑛 −

24

25
    (49) 

𝑅𝑆5(𝑃𝐷2) =
59

9
2𝑛 −

11

5
     (50) 



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Proof. Let 𝑃𝐷2i denote PAMAM dendrimers with different core unit generated by dendrimer 

𝐺𝑛 with 𝑛 growth stages. The edge set of 𝑃𝐷2i can be partitioned into four disjoint sets 

𝜀1,2,𝜀1,3,𝜀2,2 and 𝜀2,3, where 𝜀(𝑃𝐷2) = 𝜀1,2i ∪𝜀1,3 ∪𝜀2,2 ∪𝜀2,3. Further, ∣𝜀1,2∣ = 4 ⋅ 2𝑛, ∣𝜀1,3∣ =i8 ⋅ 

2𝑛− 4, ∣𝜀2,2∣ = 24 ⋅ 2𝑛− 11 and ∣𝜀2,3∣ = 28 ⋅ 2𝑛− 14. Such that ∣(𝑃𝐷2)∣ = ∣𝜀1,2∣ + ∣𝜀1,3∣ + ∣𝜀2,2∣ + 

∣𝜀2,3. Thus, with this background by employing equations (1)-(5) and (6)-(10), we get the 

required results.  

 

Theorem 11. For their PAMAM dendrimers 𝐷𝑆1, we have 

𝑅𝑆2(𝐷𝑆1) =
32

25
3𝑛 −

253

200
    (51) 

𝑅𝑆3(𝐷𝑆1) =
87

40
3𝑛 −

11

8
     (52) 

𝑅𝑆3(𝐷𝑆1) =
165

256
3𝑛 −

161

256
    (53) 

𝑅𝑆2(𝐷𝑆1) =
105

64
3𝑛 −

16

25
    (54) 

 

Proof. Let 𝐷𝑆1i denote PAMAM dendrimers with different core unit generated by dendrimer 

𝐺𝑛 with 𝑛 growth stages. The edge set of 𝐷𝑆1 can be partitioned into three 

disjoint sets 𝜀1,4,𝜀2,2 and 𝜀2,4, where 𝜀(𝐷𝑆1) = 𝜀1,4i ∪𝜀2,2 ∪𝜀2,4. Further, ∣𝜀1,4∣ = 4 ⋅ 3𝑛, ∣𝜀2,2∣ 

=10⋅3𝑛−10, and ∣𝜀2,4∣ = 4⋅3𝑛−4. Such that ∣(𝐷𝑆1)∣ = ∣𝜀1,4∣+∣𝜀2,2∣+∣𝜀2,4∣. Thus, 

with this background by employing equations (1)-(5) and (6)-(10), we get the required 

results.  

Theorem 12. For a linear polyomino chain 𝐿𝑛we have 

𝑅𝑆2(𝐿𝑛) =
1

18
𝑛 +

143

500
     (55) 

𝑅𝑆3(𝐿𝑛) =
1

18
𝑛 +

393

1000
     (56) 

𝑅𝑆4(𝐿𝑛) =
1

243
𝑛 +

31

200
     (57) 

𝑅𝑆5(𝐿𝑛) =
1

243
𝑛 +

873

500
     (58) 

 

Proof. Let 𝐿𝑛be their polyomino chain with 𝑛squares where 𝑙1 = I 𝑛and 𝑚= 1. The edge set 

of 𝐿𝑛can be partitioned into three disjoint sets 𝜀2,2,𝜀2,3 and 𝜀3,3, I where 𝜀(𝐿𝑛) = 𝜀2,2i ∪𝜀2,3 

∪𝜀3,3. Further, ∣𝜀2,2∣ = 2,i ∣𝜀2,3∣ = 4, and ∣𝜀3,3∣ = 3𝑛− 5. Such that ∣(𝐿𝑛)∣ = ∣𝜀2,2∣ + ∣𝜀2,3∣ + ∣𝜀3,3∣. 

Thus, with this background by employing equations (1)-(5) and (6)-(10), we get the required 

results. 

Theorem 13. Let 𝑍𝑛be zigzag polyomino chain with 𝑛 squares such that 𝑙= 2 and 𝑚= 𝑛− 1. 

Then 

𝑅𝑆2(𝑍𝑛) =
63

16640
𝑚 +

3

512
𝑛 +

37

100
   (59) 

𝑅𝑆3(𝑍𝑛) =
15

256
𝑚 +

3

512
𝑛 +

43

100
    (60) 

𝑅𝑆4(𝑍𝑛) =
63

32768
𝑚 +

3

65536
𝑛 +

4

25
   (61) 



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𝑅𝑆5(𝑍𝑛) =
255

32768
𝑚 +

3

65536
𝑛 +

17

100
   (62) 

 

Proof Let 𝑍𝑛be zigzag polyomino chain with 𝑛squares such that 𝑙= 2 and 𝑚= 𝑛−1. 

Polyomino chain consists of a sequence of segments 𝑆1,2,⋅⋅⋅,𝑆𝑚and 𝑙(𝑆) = 𝑙where 𝑚≥ 1 and ∈ 

{1,2,⋅⋅⋅,𝑚}. The edge set of 𝑍𝑛can be partitioned into five disjoint sets 𝜀2,2,𝜀2,3, 𝜀2,4, 𝜀3,4 

and 𝜀4,4, where 𝜀(𝑍𝑛) = 𝜀2,2i ∪𝜀2,3 ∪𝜀2,4 ∪𝜀3,4 ∪𝜀4,4. Further, ∣𝜀2,2∣ = 2, ∣𝜀2,3∣ = 4, ∣𝜀2,4∣ = 

2(𝑚− 1), ∣𝜀3,4∣ = 2 and ∣𝜀4,4∣ = 3𝑛− 2𝑚− 5. Such that ∣(𝑍𝑛)∣ =∣𝜀2,2∣ + ∣𝜀2,3∣ + ∣𝜀2,4∣ + ∣𝜀3,4∣ +i 

∣𝜀4,4∣. Thus, with this background by employing equations (1)-(5) and (6)-(10), we get the 

required results.  

Theorem 14. For the polyomino chain with 𝑛 squares and of 𝑚 segments 𝑆1 and 𝑆2i satisfying 

𝑙1 = 2 and 𝑙2 = 𝑛− 1, 𝐵𝑛
1(𝑛≥ 3) we have the following: 

𝑅𝑆2(𝐵𝑛
1) =

1

18
𝑛 +

6

25
     (63) 

𝑅𝑆3(𝐵𝑛
1) =

1

18
𝑛 +

41

100
     (64) 

𝑅𝑆4(𝐵𝑛
1) =

1

243
𝑛 +

4

25
     (65) 

𝑅𝑆5(𝐵𝑛
1) =

1

243
𝑛 +

9

50
     (66) 

Proof. Let 𝐵𝑛
1(𝑛≥ 3) be the polyomino chain with 𝑛squares and 𝑚segments 𝑆1i and 𝑆2i 

satisfying 𝑙1 = 2 and 𝑙2 = 𝑛− 1. The edge set of 𝐵𝑛
1(𝑛≥ 3) can be partitioned into five disjoint 

sets 𝜀 2,2,𝜀2,3, 𝜀2,4, 𝜀3,3i and 𝜀3,4, where𝜀(𝐵𝑛
1(𝑛 ≥ 3)) = 𝜀22 ∪ 𝜀23 ∪  𝜀24 ∪ 𝜀33 ∪ 𝜀34. Further, 

∣𝜀2,2∣ = 2, ∣𝜀2,3∣ = 5, ∣𝜀2,4∣ = 1, ∣𝜀3,3∣ = 3𝑛− 10 and ∣𝜀3,4∣ = 3. Such that ∣(𝐵𝑛
1(𝑛≥ 3)) ∣𝜀2,2∣ + ∣𝜀2,3∣ 

+ ∣𝜀2,4∣ + ∣𝜀3,3∣ + ∣𝜀3,4∣. Thus, with this background by employing equations (1)-(5) and (6)-

(10), we get the required results.  

 

Theorem 15. For their polyomino chain with 𝑛 squares and of 𝑚 segments 

𝑆1,2,⋅⋅⋅𝑠𝑚satisfying 𝑙1 = 𝑙𝑚= 2 and 𝑙2,𝑙3,⋅⋅⋅,≥ 3  𝐵𝑛
2(𝑛≥ 4) we have the following: 

𝑅𝑆2(𝐵𝑛
2) = −

2543

78957
𝑚 +

1

18
𝑛 +

29

100
   (67) 

𝑅𝑆3(𝐵𝑛
2) =

17

500
𝑚 +

1

18
𝑛 +

33

100
    (68) 

𝑅𝑆4(𝐵𝑛
2) =

1

100
𝑚 +

1

243
𝑛 +

13

100
    (69) 

𝑅𝑆5(𝐵𝑛
2) =

1

50
𝑚 +

1

243
𝑛 +

13

100
    (70) 

Proof. Let 𝐵𝑛
2(𝑛≥4) be the polyomino chain with 𝑛squares and 𝑚segments 𝑚segments 

𝑆1,𝑆2,⋅⋅⋅𝑠𝑚satisfying 𝑙1 = 𝑙𝑚= 2 and 𝑙2,𝑙3,⋅⋅⋅,≥ 3. The edge set of  𝐵𝑛
2(𝑛≥4) can be partitioned 

into five disjoint sets 𝜀2,2,𝜀2,3, 𝜀2,4, 𝜀3,3 and 𝜀3,4 where 𝜀(𝐵𝑛
2(𝑛 ≥ 3)) = 𝜀22 ∪ 𝜀23 ∪ 𝜀24 ∪

 𝜀33 ∪ 𝜀34.Further, ∣𝜀2,2∣ = 2, ∣𝜀2,3∣ = 2𝑚, ∣𝜀2,4∣ = 2, ∣𝜀3,3∣ = 3𝑛−6𝑚+3 and ∣𝜀3,4∣ = 4𝑚−6. 

Such that ∣(𝐵𝑛
2(𝑛≥ 3))∣ = ∣𝜀2,2∣+∣𝜀2,3∣+∣𝜀2,4∣+∣𝜀3,3∣+∣𝜀3,4∣.  Thus, with this background by 

employing equations (1)-(5) and (6)-(10), we get the required results.  

Theorem 16. Let 𝑇𝑝be a triangular benzenoid where 𝑝i shows the number of hexagons in the 

base graphic and total number of hexagons in 𝑇𝑝 is 
𝑝(𝑝+1)

2
. Then 



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𝑅𝑆2(𝑇𝑝) =
1

36
𝑝2 +

185

1116
𝑝 +

69

124
   (71) 

𝑅𝑆3(𝑇𝑝) =
1

36
𝑝2 +

199

612
𝑝 +

27

68
    (72) 

𝑅𝑆4(𝑇𝑝) =
1

486
𝑝2 +

13

243
𝑝 +

23

72
    (73) 

𝑅𝑆5(𝑇𝑝) =
1

486
𝑝2 +

79

972
𝑝 +

7

24
    (74) 

Proof. Let 𝑇𝑝 be a triangular benzenoid where 𝑝shows the number of hexagons in the base 

graphic and total number of hexagons in 𝑇𝑝 is 
𝑝(𝑝+1)

2
. The edge set of 𝑇𝑝 can be partitioned 

into three disjoint sets 𝜀2,2,𝜀2,3i and 𝜀3,3 where 𝜀(𝑇𝑝) = 𝜀2,2∪𝜀2,3∪𝜀3,3. Further, ∣𝜀2,2∣ = 6, ∣𝜀2,3∣ = 

6(𝑝− 1) and |𝜀33| =
3𝑝(𝑝−1)

2
 . Such that ∣(𝑇𝑝)∣ = ∣𝜀2,2∣ + ∣𝜀2,3∣ + ∣𝜀3,3∣. Thus, with this 

background by employing equations (1)-(5) and (6)-(10), we get the required results.  

 

Theorem 17. Let 𝑋𝑝 be a benzenoid hourglass. Then 

𝑅𝑆2(𝑋𝑝) =
1

18
𝑝2 +

185

558
𝑝 +

467

837
    (75) 

𝑅𝑆3(𝑋𝑝) =
1

18
𝑝2 +

199

306
𝑝 +

61

459
    (76) 

𝑅𝑆4(𝑋𝑝) =
1

243
𝑝2 +

26

243
𝑝 +

521

1458
   (77) 

𝑅𝑆5(𝑋𝑝) =
1

243
𝑝2 +

79

486
𝑝 +

413

1458
   (78) 

 

Proof. Let 𝑋𝑝be a benzenoid hourglass. The edge set of 𝑋𝑝can be partitioned into three 

disjoint sets 𝜀2,2,𝜀2,3i and 𝜀3,3i where 𝜀(𝑋𝑝) = 𝜀2,2 ∪𝜀2,3i ∪𝜀3,3. Further, ∣𝜀2,2∣ = 8, ∣𝜀2,3∣ = 

4(3𝑝−4) and ∣𝜀3,3∣ = 3𝑝2−3𝑝+4. Such that ∣(𝑋𝑝)∣ = ∣𝜀2,2∣+∣𝜀2,3∣+∣𝜀3,3∣. Thus, with this 

background by employing equations (1)-(5) and (6)-(10), we get the required results.  

Theorem 18. Let 𝐵𝑝,be denote a jagged rectangle benzenoid system for all 𝑝,𝑞∈𝑁−1. Then 

 

𝑅𝑆2(𝐵𝑝,𝑞) =
1

9
𝑝𝑞 +

247

1674
𝑝 +

959

3348
𝑞 +

497

1674
   (79) 

𝑅𝑆3(𝐵𝑝,𝑞) =
1

9
𝑝𝑞 +

233

918
𝑝 +

721

1836
𝑞 +

175

918
   (80) 

𝑅𝑆4(𝐵𝑝,𝑞) =
2

243
𝑝𝑞 +

26

729
𝑝 +

905

5832
𝑞 +

605

2916
   (81) 

𝑅𝑆5(𝐵𝑝,𝑞) =
2

243
𝑝𝑞 +

79

1458
𝑝 +

1013

5832
𝑞 +

551

2916
   (82) 

 

Proof. Let 𝐵𝑝,be denotes a jagged rectangle benzenoid system for all 𝑝,𝑞∈𝑁− 1. The edge set 

of 𝐵𝑝,𝑞can be partitioned into three disjoint sets 𝜀2,2,𝜀2,3 and 𝜀3,3 where 𝜀(𝐵𝑝,𝑞)= 𝜀2,2 ∪𝜀2,3 

∪𝜀3,3. Further, ∣𝜀2,2∣ = 2𝑞+4, ∣𝜀2,3∣ = 4𝑝+4𝑞−4 and ∣𝜀3,3∣ =6𝑝𝑞+𝑝−5𝑞−4. Such that ∣(𝐵𝑝,𝑞)∣ = 

∣𝜀2,2∣ + ∣𝜀2,3∣ + ∣𝜀3,3∣. Thus, with this background by employing equations (1)-(5) and (6)-

(10), we get the required results.  

 

 



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Conclusion 

In this paper, we have computed the topological index values of chemical structures like 

hexagonal parallelogram 𝑃(𝑚, 𝑛)  nanotube, triangular benzenoid 𝐺𝑛,zigzag-edge coronoid 

fused with starphene nanotubes 𝑍𝐶𝑆(𝑘, 𝑙, 𝑚),  dominating derived networks 𝐷1, 𝐷2, 𝐷3 , 

Porphyrin Dendrimer, Zinc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimer, 

Poly(Ethylene amido amine Dendrimer, PAMAM dendrimers(𝑃𝐷1,𝑃𝐷2,𝐷𝑆1), linear 

polyomino chain 𝐿𝑛, 𝑍𝑛, 𝐵𝑛
1(𝑛 ≥ 3), 𝐵𝑛

2(𝑛 ≥ 3)  and triangular, hourglass, and jagged-

rectangle benzenoid systems. 

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