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Neighbourhood Sum VDB Indices and Entropy for Specific Types of 

Molecular Graphs 

𝐕𝐢𝐣𝐚𝐲𝐚𝐤𝐮𝐦𝐚𝐫 𝐊∗𝟏,    𝐒. 𝐌𝐨𝐡𝐚𝐧𝟐,   𝐊. 𝐉𝐮𝐥𝐢𝐞𝐭𝐫𝐚𝐣𝐚𝟑 

1,2,3 Department of Mathematics, Presidency University, Bengaluru, Karnataka 560064, India. 

vijayakesav.k08@gmail.com , smohanmat@gmail.com, julietraja1@gmail.com 

 

Article History: 

Received: 13-11-2024 

Revised:25-12-2024 

Accepted:09-01-2025 

Abstract:  

Introduction: Numerous TIs for different molecular graphs have been found and 

studied. The entropy measurements and the neighbourhood sum degree-based M-

polynomial are derived for the six different anti-asthmatic drugs by using a set of 19 

degree-based topological indices in this article. 

Objectives: Develop the neighbourhood sum degree-based M-polynomial for different 

anti-asthmatic drugs to characterize their molecular graphs. Compare the entropy 

measurements obtained for them. 

Methods: Based on these molecular descriptors, it is practical and effective to analyse 

the mathematical values and conduct additional research on a molecule's numerous 

physical properties. They offer helpful substitutes for drawn-out, costly, and labour-

intensive laboratory investigations. Using quantitative structure-activity relationships 

(QSARs) and quantitative structure-property relationships (QSPRs), the topological 

indices can be utilized to predict chemical structures, physicochemical properties, and 

biological activities. 

Results: Based on their matching molecular structures, the topological indices of the 

six different anti-asthmatic drugs   are determined in this article. 

Conclusions: A graphical comparison of the calculated indices is made to examine 

how they relate to the molecular structure and to one another. 

Keywords: Degree-based indices, M-Polynomial, entropy measures, and 

neighborhood sum degree-based indices. 

1. Introduction 

The primary techniques for theoretically investigating chemical compounds are increasingly 

graph theoretical tools. These methods are crucial to the creation of new, more effective 

herbicides since their properties can be estimated prior to synthesis. Furthermore, QSPR/QSAR 

models can replace experimental measurements because they are less ex pensive and time-

consuming. 

In this context, topological indices provide a numerical representation of molecular topology. 

The computation of TIs can be done effectively and practically using algebraic polynomials. 

With this method, estimating several TIs is reduced to computing a single polynomial. For 

example, to find TIs based on distance, the Hosoya polynomial is widely employed. Deutsch 

and Klavˇzar created a comparable polynomial for the degree-based index calculation. By using 

this technique



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, several important degree-based benchmarks are computed using a single polynomial, known 

as the M-polynomial. Numerous scholarly works address the M-polynomial and its application 

in the computation of degree-based indices. The degree-based entropy measures are computed 

for these structures. The neighbourhood sum degree-based TIs are one recent development in 

the application of graph theory to chemical compound research. The degree-based entropy 

metrics that are widely studied and employed in graph theory are considered information 

functionals in the study of networks. Applications of entropy network measures include the 

study of the chemical and biological properties of molecular graphs and the quantitative 

description of a molecule’s structure. We consider a simple connected graph that has few edges 

and no self-loops. 

The graph 𝔊 is referred to as a connected graph when 𝑉(𝔊) and 𝐸(𝔊) represent the vertex set 

and edge set, respectively. The degree of the vertex is indicated by the symbol 𝒹𝓋.  

The neighbourhood sum degree-based TI's are denoted by 𝔑𝔊(𝔳). The neighbourhood sum 

degree of the molecular graph is written as |𝕹𝕲(𝖛)| = 𝓭𝓿. The 𝕹𝕲(𝖛) denotes the sum of the 

degrees of vertices that are next to 𝖛. Let’s start by building the M-polynomial of 𝕲 based on 

the neighbourhood sum degree: 

𝔑𝔐(𝔊) =  𝛴(i≤j)(Number of all edges 𝔲𝔳 such that H𝔲 = i, H𝔳 = j)𝑙𝑖𝑝𝑗 

𝔇(𝔊) = Σ𝔲𝔳∈𝔼(𝔊)𝑔(𝜑𝑢𝜑𝑣) 𝑎𝑛𝑑 𝔑𝔐(𝔊) = Σ𝔲𝔳∈𝔼(𝔊)𝑔(𝔴𝑢𝔴𝑣)  

 

1 𝑀1(𝔊) = 𝐷𝑛 + 𝐷𝑣(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) First Zagreb Index 

2 𝑀2(𝔊) = 𝐷𝑛 ∗  𝐷𝑣(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) Second Zagreb Index 

3 𝑀2
𝑚(𝔊) = 𝑆𝑛 ∗ 𝑆𝑣(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) Second modified Zagreb Index 

4 𝐷𝑛
𝛼 ∗  𝐷𝑣

𝛼(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) General Randic Index 

5 𝑆𝑛
𝛼 ∗  𝑆𝑣

𝛼(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1)  Inverse Randic Index 

6 [𝐷𝑛𝑆𝑣 + 𝐷𝑛𝑆𝑣](𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) Symmetric Division Index 

7 2𝑆𝑛𝐽(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=1) Harmonic Index 

8 𝑆𝑛𝐽𝐷𝑛 𝐷𝑣(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=1) Inverse Sum Index 

9 𝕊s
3Q−2 J𝔻s

3 𝔻p
3(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=1) Augumented Redefined Zagreb 

Index  

10 [𝐷𝑛 + 𝐷𝑣][𝑆𝑛. 𝑆𝑣]−1(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) First Redefined Zagreb Index 

11 [𝐷𝑛 ∗ 𝐷𝑣][𝐷𝑛 + 𝐷𝑣](𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1)  Third Redefined Zagreb Index 

12 𝐷𝑛
1/2

𝑄−2𝐽[𝑆𝑛
1/2

. 𝑆𝑛
1/2

](𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=1) Atom–bond sum Index 



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13 𝐷𝑛
1/2

𝑄−2𝐽[𝑆𝑛
1/2

+ 𝑆𝑛
1/2

](𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=1) Atom–bond Connectivity 

Index 

14 [𝐷𝑛
2 + 𝐷𝑛

2](𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1)  F-Index 

15 1

4
[𝐷𝑔+𝐷ℎ]2(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) 

 Second Shigehalli and 

Kanaburs Index 

16 [𝐷𝑛
1/2

∗  𝐷𝑛
1/2

](𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1)  Reciprocal Randic Index 

17 [𝐷𝑔 + 𝐷ℎ]−1(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) General Sum-connectivity 

Index  

18 1

2
[𝐷𝑛 ∗ 𝐷𝑣][𝐷𝑛 + 𝐷𝑣](−

1
2

)(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) 
Arithmetic Geometric Index 

19 
2[𝐷𝑛𝐷𝑣]

1
2[𝐷𝑛 + 𝐷𝑣](−1)(𝔑𝔐(𝔊: 𝑔, ℎ))|(𝑔=ℎ=1) 

Geometric Arithmetic Index  

 

Table1aboveliststheNM-Polynomialderivatives[11,10,17,1] 

2. Neighbourhood Degree Sum-Based Entropy Measures: 

In his seminal work, Shannon defined entropy as a means of quantifying the degree of 

uncertainty in a system or the unexpectedness of relevant information. The structural 

informativeness of a network has been measured using entropy computations [18]. Information 

theory has been widely applied because of its versatility in many different domains, including 

linguistics, electrical engineering, chemical and medical sciences, and graph theory for 

chemical networks[6,5]. A method for measuring the topological information of chemical 

networks and graphs was introduced graph entropy. 

[19, 16] Rashevsky computed graph entropy using vertex orbits. Using intrinsic and extrinsic 

graph entropy measurements, mathematicians may relate probability distributions to graph 

elements such as vertices and edges. Graph entropies are widely used in many fields, including 

as biology, ecology, chemistry, and sociology. [5, 7] Dehmer created graph entropies and used 

information functional analysis to extract structural information from them. 𝐸𝑁𝑇𝔊 =

Σ𝑗=1
𝑏𝑖  𝑝𝑖𝑗𝑙𝑜𝑔𝑝. If 𝔊 = (𝑉, 𝐸, 𝑤) is an edge-weighted graph then the entropy measure of G is 

defined as [4, 8]. 

𝚬𝚴𝚻𝕲 = 𝒍𝒐𝒈𝛀(𝕲) −
𝟏

𝛀(𝕲)
𝚺𝖚𝖛∈𝔼(𝕲)𝑭(𝖉𝒖𝖉𝒗)𝒍𝒐𝒈(𝑭(𝖉𝒖𝖉𝒗)) 

3.Computing Neighbourhood Sum Degree-Based M - polynomial by Using TI’s. 

We compute the NM-Polynomials for isoproterenol, terbutaline, pranlukast, setipiprant, 

bedoradrine and toreforant. These drugs are often used as part of a comprehensive treatment 

plan for asthma and COPD, either alone or in combination with other medications such as 

inhaled corticosteroids or short-acting beta agonists. It’s important to use them as prescribed 



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by a healthcare professional and to be aware of their potential side effects and interactions. The 

below graphs are indicating their molecular graphs. 

 

a) Isoproterenol                                                                        b)  Terbutaline 

 

  

c) Pranlukast                                                                 d) Setipiprant 



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e) Bedoradrine                                                              f) Toreforant   

 

                                  Figure.1. Anti-Asthmatic-Drugs 

 

( 𝖉𝖚,, 𝖉𝖛)  Frequency 

(3,5) 2 

(3,6) 1 

(4,5) 3 

(5,6) 7 

(6,7) 3 

Table 3: Partition table for terbutaline 

Table 2: Partition table for isoproterenol 

 

( 𝖉𝖚,, 𝖉𝖛) Frequency 

3,6) 2 

(4,4) 5 

(4,5) 8 

(5,5) 2 

(5,6) 6 

(5,7) 5 

( 𝖉𝖚,, 𝖉𝖛) Frequency 

(3,4) 2 

(3,6) 3 

(4,5) 1 

(5,5) 2 

(5,6) 2 

(5,7) 1 

(6,6) 2 

(6,7) 2 

 

( 𝖉𝖚,, 𝖉𝖛) Frequency 

(3,4) 1 

(3,5) 2 

(3,7) 1 

(4,5) 5 

(4,6) 1 

(5,5) 3 

(5,6) 1 

(5,7) 3 

(5,8) 4 

(6,7) 1 

(6,8) 3 

(7,7) 1 

(7,8) 2 

(8,8) 6 

 



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2 

(5,8) 1 

(6,6) 2 

(6,7) 4 

(6,8) 2 

(7,7) 1 

(7,8) 1 

(8,8) 1 

Table 4: Partition table for pranlukast Table 5: Partition table for setipiprant 

( 𝖉𝖚,, 𝖉𝖛) Frequency 

(2,3) 1 

(3,5) 3 

(3,6) 3 

(5,5) 5 

(5,6) 8 

(5,7) 4 

(6,6) 2 

(6,7) 6 

(7,7) 1 

Table 6: Partition table for bedoradrine 

                                                                                               Table 7: Partition table for 

toreforant 

 

Theorem 3.1. Given that 𝔊 to be a isoproterenol, the following 𝔑𝔐 -polynomial of 

𝔊. 𝔑𝔐(𝔊, g, h) = 2g3h4 + 3g3h6 + g4h5 + 2g5h5 + 2g5h6 + g5h7 + 2g6h6 + 2g6h7 

Theorem 3.2. If 𝔊 is a isoproterenol, then that 𝔊 ′s NM -polynomial can be found. 

1. 𝔑𝔐M1
(𝔊) = 154 

2. 𝔑𝔐M2
(𝔊) = 399 

3. 𝔑𝔐M2
m(𝔊) = 0.66175 

( 𝖉𝖚,, 𝖉𝖛) Frequency 

(3,5) 2 

(3,6) 2 

(4,4) 1 

(4,5) 2 

(5,5) 5 

(5,6) 7 

(5,8) 1 

(6,6) 3 

(6,7) 4 

(6,8) 5 

 



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2 

4. 𝔑𝔐GRI(𝔊) = 399 

5. 𝔑𝔐IRI(𝔊) = 0.66175 

6. 𝔑𝔐SDI(𝔊) = 31.945 

7. 𝔑𝔐HI(𝔊)= 3.0316 

8. 𝔑𝔐ISI(𝔊) = 37.484 

9. 𝔑𝔐AZ(𝔊) = 484.6 

10. 𝔑𝔐ReZ1
(𝔊) = 6.2786 

11. 𝔑𝔐ReZ3
(𝔊)= 4370 

12. 𝔑𝔐ABC(𝔊)= 8.5924 

13. 𝔑𝔐ABS(𝔊) = 13.394 

14. 𝔑𝔐F(𝔊) = 836 

15. 𝔑𝔐SK2
(𝔊) = 408.5 

16. 𝔑𝔐RR(𝔊) = 75.959 

17. 𝔑𝔐SI(𝔊)= 4.7455 

18. 𝔑𝔐GA1
(𝔊) = 15.237 

19. 𝔑𝔐AG(𝔊)= 14.774 

Theorem 3.3. Given that 𝔊 to be a terbutaline, the following 𝔑𝔐 -polynomial of 𝔊. 

𝔑𝔐(𝔊, g, h) = 2g3h5 + g3h6 + 3g4h5 + 7g5h6 + 3g6h7 

Theorem 3.4. If 𝔊 is a terbutaline, then that 𝔊 ′s 𝔑𝔐 -polynomial can be found. 

1. 𝔑𝔐M1
(𝔊) = 168 

2. 𝔑𝔐M2
(𝔊) = 444   

3. 𝔑𝔐M2
m(𝔊) =  0.64365 

4. 𝔑𝔐GRI(𝔊) = 444 

5. 𝔑𝔐IRI(𝔊) = 0 .64365  

6. 𝔑𝔐SDI(𝔊) = 33.488 

7. 𝔑𝔐HI(𝔊)= 3.1232 

8. 𝔑𝔐ISI(𝔊) = 41 .2  

9. 𝔑𝔐AZ(𝔊) = 5 4 4 . 4 7  

10. 𝔑𝔐ReZ1
(𝔊) = 6.4119 

11. 𝔑𝔐ReZ3
(𝔊)= 4 8 9 0  

12. 𝔑𝔐ABC(𝔊)= 9 . 0 2 3 6  



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2 

13. 𝔑𝔐ABS(𝔊) = 1 4 . 3 5  

14. 𝔑𝔐F(𝔊) = 918 

15. 𝔑𝔐SK2
(𝔊) = 451.5  

16. 𝔑𝔐RR(𝔊) = 8 3 . 1 8 8  

17. 𝔑𝔐SI(𝔊)= 4.9831 

18. 𝔑𝔐GA1
(𝔊) = 16.183 

19. 𝔑𝔐AG(𝔊)= 1 5 . 8 2 3  

Theorem 3.5. Given that 𝔊 to be a pranlukast, the following 𝔑𝔐 -polynomial of 𝔊. 

𝔑𝔐 (𝔊; g, h) = 2g3h6 + 5g4h4 + 8g4h5 + 2g5h5 + 6g5h6 + 5g5h7 + g5h8 + 2g6h6 + 4g6h7 

+ 2g6h8 + g7h7 + g7h8 + g8h8 

Theorem 3.6. If 𝔊 is a pranlukast, then that 𝔊 ′s 𝔑𝔐 -polynomial can be found. 

1. 𝔑𝔐M1
(𝔊)= 438 

2. 𝔑𝔐M2
(𝔊) = 1226 

3. 𝔑𝔐M2
m(𝔊) = 1.5178 

4. 𝔑𝔐GRI(𝔊) = 1226 

5.  𝔑𝔐IRI(𝔊) =1.5178 

6. 𝔑𝔐SDI(𝔊) = 82.676 

7. 𝔑𝔐HI(𝔊) = 7.3359 

8. 𝔑𝔐ISI(𝔊) = 107.82 

9. 𝔑𝔐AZ(𝔊) = 1548.6 

10. 𝔑𝔐ReZ1
(𝔊) = 15.431 

11. 𝔑𝔐ReZ3
(𝔊)= 14446 

12. 𝔑𝔐ABC(𝔊) = 22.201 



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2 

13. 𝔑𝔐ABS(𝔊) = 35.818 

14. 𝔑𝔐F(𝔊) = 2526 

15. 𝔑𝔐SK2
(𝔊) = 1244.5 

16. 𝔑𝔐RR(𝔊)= 217.28 

17. 𝔑𝔐SI(𝔊) = 12.203 

18. 𝔑𝔐GA1
(𝔊)= 40.33 

19. 𝔑𝔐AG(𝔊)= 39.681 

Theorem 3.7. Given that 𝔊 to be a setipiprant, the following 𝔑𝔐 -polynomial of  𝔊 . 

𝔑𝔐 (𝔊; g, h) = g3h4 + 2g3h5 + g3h7 + 5g4h5 + g4h6 + 3g5h5 + g5h6 + 3g5h7 + 4g5h8 + 

g6h7 + 3g6h8 + g7h7 + 2g7h8 + 6g8h8 

Theorem 3.8. If 𝔊 is a setipiprant, then that 𝔊 ′s 𝔑𝔐 -polynomial can be found. 

1. 𝔑𝔐M1
(𝔊)= 412 

2. 𝔑𝔐M2
(𝔊)  = 1288 

3. 𝔑𝔐M2
m(𝔊) = 1.1312 

4. 𝔑𝔐GRI(𝔊) = 1288 

5. 𝔑𝔐IRI(𝔊)  = 1.1312 

6. 𝔑𝔐SDI(𝔊) = 71.381 

7. 𝔑𝔐HI(𝔊) = 5.936 

8. 𝔑𝔐ISI(𝔊) = 100.84 

9. 𝔑𝔐AZ(𝔊)  = 1705.1 

10. 𝔑𝔐ReZ1
(𝔊) = 12.194 

11. 𝔑𝔐ReZ3
(𝔊) = 17166 

12. 𝔑𝔐ABC(𝔊) = 18.287 

13. 𝔑𝔐ABS(𝔊)  = 30.88 

14. 𝔑𝔐F(𝔊) = 2641 

15. 𝔑𝔐SK2
(𝔊)= 1312.5 

16. 𝔑𝔐RR(𝔊)= 203.82 

17. 𝔑𝔐SI(𝔊)=  9.9728 

18. 𝔑𝔐GA1
(𝔊)= 34.416 

19. 𝔑𝔐AG(𝔊)= 33.6 

 



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2 

2 

Theorem 3.9. Given that 𝔊 to be a bedoradrine , the following 𝔑𝔐 -polynomial of 

𝔊. 𝔑𝔐 (G; g, h) = g2h3 + 3g3h5 + 3g3h6 + 5g5h5 + 8g5h6 + 4g5h7 + 2g6h6 + 6g6h7 + 

g7h7 

Theorem 3.10. If 𝔊 is a bedoradrine, then that 𝔊 ′s 𝔑𝔐 -polynomial can be found. 

1. 𝔑𝔐M1
(𝔊) = 358 

2. 𝔑𝔐M2
(𝔊) = 983 

3. 𝔑𝔐M2
m(𝔊) = 1.3331 

4. 𝔑𝔐GRI(𝔊) = 983  

5. 𝔑𝔐IRI(𝔊) = 1.3331 

6. 𝔑𝔐SDI(𝔊)  = 69.333 

7. 𝔑𝔐HI(𝔊)= 6.3371 

8. 𝔑𝔐ISI(𝔊)= 87.694 

9. 𝔑𝔐AZ(𝔊)  = 1221.6 

10. 𝔑𝔐ReZ1
(𝔊)  = 13.048 

11. 𝔑𝔐ReZ3
(𝔊)= 11272 

12. 𝔑𝔐ABC(𝔊)= 18.444 

13. 𝔑𝔐ABS(𝔊) = 29.649 

14. 𝔑𝔐F(𝔊)= 2036 

15. 𝔑𝔐SK2
(𝔊)= 1000.5 

16. 𝔑𝔐RR(𝔊)= 177.15 

17. 𝔑𝔐SI(𝔊)= 10.165 

18. 𝔑𝔐GA1
(𝔊)  = 33.409 

19. 𝔑𝔐AG(𝔊) = 32.606 

Theorem 3.11. Given that 𝔊 to be a toreforant, the following 𝔑𝔐 -polynomial of 𝔊. 

𝔑𝔐 (𝔊; g, h) = 2g3h5 + 2g3h6 + g4h4 + 2g4h5 + 5g5h5 + 7g5h6 + g5h8 + 3g6h6 + 4g6h7 

+ 3g6h8 + 2g7h8 

Theorem 3.12. If 𝔊 is a toreforant, then that 𝔊 ′s 𝔑𝔐 -polynomial can be found. 

1. 𝔑𝔐M1
(𝔊)= 360 

2. 𝔑𝔐M2
(𝔊) = 1029 

3. 𝔑𝔐M2
m(𝔊) = 1.1421 

4. 𝔑𝔐GRI(𝔊) = 1029 



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5. 𝔑𝔐IRI(𝔊) = 1.1421 

6. 𝔑𝔐SDI(𝔊) = 66.473 

7. 𝔑𝔐HI(𝔊) = 5.8761 

8. 𝔑𝔐ISI(𝔊)= 88.538 

9. 𝔑𝔐AZ(𝔊)  = 1305.3 

10. 𝔑𝔐ReZ1
(𝔊)  = 12.007 

11. 𝔑𝔐ReZ3
(𝔊)= 12308 

12. 𝔑𝔐ABC(𝔊) = 17.586 

13. 𝔑𝔐ABS(𝔊)= 28.907 

14. 𝔑𝔐F(𝔊)= 2120 

15. 𝔑𝔐SK2
(𝔊)= 1044.5 

16. 𝔑𝔐RR(𝔊)= 178.51 

17. 𝔑𝔐SI(𝔊)= 9.6568 

18. 𝔑𝔐GA1
(𝔊) = 32.304 

19. 𝔑𝔐AG(𝔊) = 31.707 

The aforementioned findings are obtained using the partition table-2 and the conditions 

of the 𝔑𝔐 -Polynomial with its derivatives[11, 8, 20, 15]. 

4. Neighbourhood Degree Sum-Based Entropy Measures of anti-asthmatic drugs . 

Theorem 4.1. If 𝔊 is a isoproterenol, then 𝔑𝔐 - measures of entropy are given as follows 

[1, 13, 14]. 

1. 𝔑𝔐ENTM1 = 2.691 

2. 𝔑𝔐ENTM2 = 2.6377 

3. 𝔑𝔐ENTM2
m= 2.6223 

4. 𝔑𝔐ENTGRI = 2.6377 

5. 𝔑𝔐ENTIRI = 2.6223 

6. 𝔑𝔐ENTSDD = 2.7043 

7. 𝔑𝔐ENTHI = 2.6885 

8. 𝔑𝔐ENTISI = 2.6868 

9. 𝔑𝔐ENTAZ = 4.4163 

10. 𝔑𝔐ENTReZ1
= 2.6848 

11. 𝔑𝔐ENTReZ3 = 2.5696 



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12. 𝔑𝔐ENTABC = 2.7047 

13. 𝔑𝔐ENTABS = 2.3269 

14. 𝔑𝔐ENTF = 2.6481 

15. 𝔑𝔐ENTSK2 = 5.7008  

16. 𝔑𝔐ENTRR= 2.6889 

17. 𝔑𝔐ENTSI= 2.7033 

18. 𝔑𝔐ENTAG1 = 2.7078 

19. 𝔑𝔐ENTGA= 2.7079 

 

Theorem 4.2. If 𝔊 is a terbutaline then 𝔑𝔐 - measures of entropy are given as follows 

[1, 13, 14]. 

1. 𝔑𝔐ENTM1 = 2.7735 

2. 𝔑𝔐ENTM2 = 2.7214 

3. 𝔑𝔐ENTM2
m= 2.7158 

4. 𝔑𝔐ENTGRINMENT GRI = 2.7214 

5. 𝔑𝔐ENTIRI = 2.7158 

6. 𝔑𝔐ENTSDD = 2.7708 

7. 𝔑𝔐ENTHI = 2.7602 

8. 𝔑𝔐ENTISI = 2.7576 

9. 𝔑𝔐ENTAZ = 4.5119 

10. 𝔑𝔐ENTReZ1
= 2.7564 

11. 𝔑𝔐ENTReZ3 = 2.6673 

12. 𝔑𝔐ENTABC = 2.7701 

13. 𝔑𝔐ENTABS = 2.7724 

14. 𝔑𝔐ENTF = 2.7306 

15. 𝔑𝔐ENTSK2 = 3.224 

16. 𝔑𝔐ENTRR= 2.7591 

17. 𝔑𝔐ENTSI= 2.7696 

18. 𝔑𝔐ENTAG1 = 2.7724 

19. 𝔑𝔐ENTGA= 2.7725 



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2 

Theorem 4.3. If  𝔊  is a , pranlukast then 𝔑𝔐 - measures of entropy are given as follows 

[1, 13, 14]. 

1. 𝔑𝔐ENTM1 = 3.6702  

2. 𝔑𝔐ENTM2 = 3.6137 

3. 𝔑𝔐ENTM2
m= 3.617 

4. 𝔑𝔐ENTGRI = 3.6137 

5. 𝔑𝔐ENTIRI =3.617 

6. 𝔑𝔐ENTSDD = 3.6876 

7. 𝔑𝔐ENTHI = 3.6929 

8. 𝔑𝔐ENTISI = 3.6693 

9. 𝔑𝔐ENTAZ = 5.2892 

10. 𝔑𝔐ENTReZ1
= 3.6698 

11. 𝔑𝔐ENTReZ3 = 3.5298 

12. 𝔑𝔐ENTABC = 3.6857 

13. 𝔑𝔐ENTABS = 3.6842 

14. 𝔑𝔐ENTF = 3.6162 

15. 𝔑𝔐ENTSK2 =4.009 

16. 𝔑𝔐ENTRR= 3.6696 

17. 𝔑𝔐ENTSI= 3.6849 

18. 𝔑𝔐ENTAG1 = 3.6894 

19. 𝔑𝔐ENTGA= 3.6882 

Theorem 4.4. If  𝔊  is a setipiprant then 𝔑𝔐 - measures of entropy are given as follows 

[1, 13, 14]. 

1. 𝔑𝔐ENTM1 = 3.5 

2. 𝔑𝔐ENTM2 = 3.4258 

3. 𝔑𝔐ENTM2
m= 3.4155 

4. 𝔑𝔐ENTGRI = 3.4258 

5. 𝔑𝔐ENTIRI =3.4155 

6. 𝔑𝔐ENTSDD = 3.5311 

7. 𝔑𝔐ENTHI = 3.4969 

8. 𝔑𝔐ENTISI = 3.4971 



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9. 𝔑𝔐ENTAZ = 4.8422 

10. 𝔑𝔐ENTReZ1
= 3.5034 

11. 𝔑𝔐ENTReZ3 = 3.3279 

12. 𝔑𝔐ENTABC = 3.5254 

13. 𝔑𝔐ENTABS = 3.5259 

14. 𝔑𝔐ENTF = 3.3666 

15. 𝔑𝔐ENTSK2 =6.8929 

16. 𝔑𝔐ENTRR= 3.499 

17. 𝔑𝔐ENTSI= 3.519 

18. 𝔑𝔐ENTAG1 = 3.5282 

19. 𝔑𝔐ENTGA= 3.5243 

Theorem 4.5. If  𝔊  is a bedoradrine then 𝔑𝔐 - measures of entropy are given as follows 

[1, 13, 14]. 

1. 𝔑𝔐ENTM1 = 3.5035 

2. 𝔑𝔐ENTM2 = 3.4362 

3. 𝔑𝔐ENTM2
m= 3.3568 

4. 𝔑𝔐ENTGRI = 3.4362 

5. 𝔑𝔐ENTIRI =3.3568 

6. 𝔑𝔐ENTSDD = 3.494 

7. 𝔑𝔐ENTHI = 3.4711 

8. 𝔑𝔐ENTISI = 3.4767 

9. 𝔑𝔐ENTAZ = 5.205 

10. 𝔑𝔐ENTReZ1
= 3.4659 

11. 𝔑𝔐ENTReZ3 = 3.3837 

12. 𝔑𝔐ENTABC = 3.493 

13. 𝔑𝔐ENTABS = 3.4959 

14. 𝔑𝔐ENTF = 3.4457 

15. 𝔑𝔐ENTSK2 =3.8808 

16. 𝔑𝔐ENTRR= 3.4783 

17. 𝔑𝔐ENTSI= 3.491 

18. 𝔑𝔐ENTAG1
= 3.4964 



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19. 𝔑𝔐ENTGA= 3.4963 

Theorem 4.6. If 𝔊 is a toreforant then 𝔑𝔐 - measures of entropy are given as follows 

[1, 13, 14]. 

1. 𝔑𝔐ENTM1 = 3.7278 

2. 𝔑𝔐ENTM2 = 3.8596 

3. 𝔑𝔐ENTM2
m= 3.1607 

4. 𝔑𝔐ENTGRI = 3.8596 

5. 𝔑𝔐ENTIRI =3.1607 

6. 𝔑𝔐ENTSDD = 3.5283 

7. 𝔑𝔐ENTHI = 3.3022 

8. 𝔑𝔐ENTISI = 3.5762 

9. 𝔑𝔐ENTAZ = 5.3788 

10. 𝔑𝔐ENTReZ1
= 3.3556 

11. 𝔑𝔐ENTReZ3 = 4.1708 

12. 𝔑𝔐ENTABC = 3.4042 

13. 𝔑𝔐ENTABS = 3.4575 

14. 𝔑𝔐ENTF = 3.9415 

15. 𝔑𝔐ENTSK2 = 4.2029 

16. 𝔑𝔐ENTRR= 3.65223 

17. 𝔑𝔐ENTSI= 3.3512 

18. 𝔑𝔐ENTAG1 = 3.4658 

19. 𝔑𝔐ENTGA= 3.465



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4. Comparative Analysis 

The three-dimensional graphs derived from Theorems 3 and 4 that illustrate the analytical 

equations for the neighbourhood degree sum-based indices are shown in this section. The 

reader will find it easier to comprehend and grasp how the indices operate in relation to 

the variables that form the molecular structure thanks to these illustrations. The differences 

between the indices and the chemical structure are graphically represented in these 

comparison charts. Tables 8–11 display the calculated numerical values for the indices. 

To aid readers in understanding the numerical data, Figure 3 and 4 present them as three-

dimensional graphs. 

 

Drugs 𝔑𝔐M1 𝔑𝔐M2 𝔑𝔐m 

2 

M 

𝔑𝔐GRI 𝔑𝔐IRI 𝔑𝔐SDD 𝔑𝔐HI 𝔑𝔐ISI 𝔑𝔐AZ 

Drug 

1 

154 399 0.66175 399 0.66175 31.945 3.0316 37.484 484.6 

Drug 

2 

168 444 0.64365 444 0.64365 33.488 3.1232 41.2 544.47 

Drug 

3 

438 1226 1.5178 1226 1.5178 82.676 7.3359 107.82 1548.6 

Drug 

4 

412 1288 1.1312 1288 1.1312 71.381 5.936 100.84 1705.1 

Drug 

5 

358 983 1.3331 983 1.3331 69.333 6.3371 87.694 1221.6 

Drug 

6 

360 1029 1.1421 1029 1.1421 66.473 5.8761 88.538 1305.3 

 

Table 8 . Comparison Table for 𝔑𝔐 -values 

 

𝔑𝔐ReZ

1 

𝔑𝔐Rez

3 

𝔑𝔐AB

C  

𝔑𝔐AB

S 

𝔑𝔐

F 

𝔑𝔐SK

2 

𝔑𝔐R

R 

𝔑𝔐

χ 

𝔑𝔐AG

1 

𝔑𝔐G

A 

6.2786 4370 8.5924 13.394 836 408.5 75.959 4.745

5 

15.237 14.774 

6.4119 4890 9.0326 14.35 918 451.5 83.188 4.983

1 

16.183 15.823 

15.431 14446 22.201 35.818 2526 1244.5 217.28 12.20 40.33 39.681 



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3 

12.194 17166 18.287 30.88 2641 1312.5 203.82 9.972

8 

34.416 33.6 

13.048 11272 18.444 29.649 2036 1000.5 177.15 10.16

5 

33.409 32.606 

12.007 12308 17.586 28.907 2120 1044.5 178.51 9.656

8 

32.304 31.707 

Table 9. Comparison Table for 𝔑𝔐 -values 

 

Drugs 𝔑𝔐M1 𝔑𝔐M2 𝔑𝔐m 

2 

M 

𝔑𝔐GRI 𝔑𝔐IRI 𝔑𝔐SDD 𝔑𝔐HI 𝔑𝔐ISI 𝔑𝔐AZ 

Drug 1 2.691 2.6377 2.6223 2.6377 2.6223 2.7043 2.6885 2.6868 4.4163 

Drug 2 2.7735 2.7214 2.7158 2.7214 2.7158 2.7708 2.7602 2.7576 4.5119 

Drug 3 3.6702 3.6137 3.617 3.6137 3.617 3.6876 3.6929 3.6693 5.2892 

Drug 4 3.5 3.4258 3.4155 3.4258 3.4155 3.5311 3.4969 3.4971 4.8422 

Drug 5 3.5035 3.4362 3.3568 3.4362 3.3568 3.494 3.4711 3.4767 5.205 

Drug 6 3.7278 3.8596 3.1607 3.8596 3.1607 3.5283 3.3022 3.5762 5.3788 

Table 10. Comparison Table for Entropy-values 

 

𝔑𝔐ReZ

1 

𝔑𝔐Rez

3 

𝔑𝔐AB

C  

𝔑𝔐AB

S 

𝔑𝔐

F 

𝔑𝔐SK

2 

𝔑𝔐R

R 

𝔑𝔐

χ 

𝔑𝔐AG

1 

𝔑𝔐G

A 

2.6848 2.5696 2.7047 2.3269 2.648

1 

5.7008 2.6889 2.703

3 

2.7078 2.7079 

2.7564 2.6673 2.7701 2.7724 2.730

6 

3.224 2.7591 2.769

6 

2.7724 2.7725 

3.6698 3.5298 3.6857 3.6842 3.616

2 

4.009 3.6696 3.684

9 

3.6894 3.6882 

3.5034 3.3279 3.5254 3.5259 3.366

6 

6.8929 3.499 3.519 3.5282 3.5243 

3.4659 3.3837 3.493 3.4959 3.445

7 

3.8808 3.4783 3.491 3.4964 3.4963 

3.3556 4.1708 3.4042 3.4575 3.941 4.2029 3.6522 3.351 3.4658 3.4653 



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5 2 

 

Table 11. Comparison Table for Entropy-values 

 

Figure 2: 3D plots for Table 8 and 9 

 

Figure 3: 3D plots for Table 10 and 1 1  

5. Conclusions 

This article computes the M-Polynomial to get the closed-form analytical formulas for the 

neighbourhood sum degree-based indices for the 6 different anti-asthmatic drugs such as 

isoproterenol, terbutaline, pranlukast, setipiprant, bedoradrine, toreforant using a set of 19 

degree - based topological indices. The results are shown as separate three-dimensional plots 

and comparison plots to aid in the comprehension of the mathematical expressions. For all the 

above drugs, Neighbourhood Degree Sum-Based Entropy Measures are computed. This 

finding will provide new perspectives for future research on topological indices for these drugs 

Funding 

This research received no specific grant from public, commercial, or not-for-profit 

funding agencies. 

Acknowledgments 

The authors wish to thank the management of Presidency University, Bengaluru, Karnataka 

560064, India., for their continuous support and encouragement to carry out this research 

work. 



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Conflicts of Interest 

The authors declare that none of the work reported in this study could have been influenced 

by any known competing financial interests or personal relationships. 

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