Journal of Mathematics
Volume 2021 (2021), Article ID 7483508, 14 pages
https://doi.org/10.1155/2021/7483508

Research Article

Computing SS Index of Certain Dendrimers

Weidong Zhao,¹ M.C. Shanmukha,² A. Usha,³ Mohammad Reza Farahani,⁴ and K.C. Shilpa⁵

¹School of Computer Science, Chengdu University, Chengdu, China
²Department of Mathematics, Jain Institute of Technology, Davanagere-577003, Karnataka, India
³Department of Mathematics, Alliance School of Applied Mathematics, Alliance University, Bangalore-562106, Karnataka, India
⁴Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran
⁵Department of Computer Science & Engg, Bapuji Institute of Engineering and Technology, Davanagere-577004, Karnataka, India

Correspondence should be addressed to M.C. Shanmukha; mcshanmukha@gmail.com

Received 25 June 2021; Accepted 28 August 2021; Published 25 September 2021

Academic Editor: Huseyin Isik

Copyright © 2021 Weidong Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The numerical descriptor gathers the data from the molecular graphs and helps to know the characteristics of the chemical structure known as topological index. The QSAR/QSPR/QSTR studies are benefited with the significant role played by topological indices in the drug design. Topological indices provide the information about the physical/chemical/biological properties of chemical compounds. The Zagreb indices are widely studied because of their extensive usage in chemical graph theory. Inspired by the earlier work on inverse sum indeg index (ISI index), novel topological index known as SS index is introduced and computed for four dendrimer structures. Also, the strong correlation coefficient between SS index and 5 physico-chemical characteristics such as boiling point (bp), molar volume (mv), molar refraction (mr), heats of vaporization (hv), and critical pressure (cp) of 67 alkane isomers have been determined. It is found that newly introduced index has shown good correlation in comparison with three most popular existing indices (ISI index and first and second Zagreb indices). In the last part, the mathematical properties of SS index are discussed.

1. Introduction and Terminologies

Every year, large number of new drugs are produced due to the rapid growth of medicine manufacturing. As a result, determining the pharmacological, chemical, and biological characteristics of a substance necessitates a significant amount of effort.

These new medications are becoming increasingly clumsy and clumped. In order to check the performance of new drugs and their side effects, sufficient reagents, equipment, and technicians are needed. However, in low-income countries, there is an insufficient funding to cover the costs of reagents and equipment needed to calculate biochemical properties. The existing studies have shown that the chemical and pharmacodynamic properties of drugs, as well as their molecular structures, are inextricably related. If we quantify measures of these drug molecular structures with the aim of identifying topological indices, medical and pharmaceutical researchers will be able to understand their therapeutic properties, which can compensate for the shortcomings of medicine and chemical experiments. In this regard, the methods computing topological index are suitable and useful for developing countries, as they can produce accessible biological and medical knowledge about new drugs without the use of chemical experiment hardware. To calculate the characteristics of drug molecules, the PI index, Zagreb index, and eccentric index are used. The number of vertices and edges of a chemical compound counts to the computation of topological indices [1–8].

A topological index is a computational parameter derived from the graph structure mathematically [9–13]. To visualize the relationships between the data sets, graphs are crucial tools which make the concept better understandable. A descriptor that gives the data regarding arrangement of atoms in a compound in numerical form of information regarding its shape, branching, and other data of a compound is a topological index.

The significant number of early drug studies suggests that the biomedical and pharmacology properties of drugs, as well as their molecular structures, have a clear inner relationship. Many scientists have developed various indices to quantify the characteristics of drug molecules over the last 40 years. The indices are of great use in the study of pharmacology, toxicology, and chemistry (QSAR/QSPR/QSTR) [14–16].

Dendrimers are also called “cascade molecules,” but this term is not in general use compared to the term dendrimers. In 1978, Fritz Vogtle was the first to bring these nanomolecules into light. Dendrites normally include a unique chemically addressable unit known as focus or core. The usage and popularity of dendrimers have been greatly increased. Since 2005, there have been over 5000 research papers and patents. A second group of the synthesized macromolecules is called arborols. We can say that the molecules of dendrimers are of architectural design. These thoroughly tailored architectural nanomolecules can be functionalized and modify their physico-chemical or biological characteristics.

The hyper-branched macromolecules have three phases in its structural constitution. An atom at the centre of the structure called the core of the dendrimer has some functional properties. Secondly, the branches are ejected out of the core and add on the branches repetitively. Finally, the terminal groups are situated on the surface of the dendritic structure. Dendrimer synthesis is divided into two methods: divergent synthesis and convergent synthesis. It is difficult to synthesize dendrimers using either approach, because the actual reactions require several steps to protect the active site. As a result, it is difficult to manufacture and prohibitively costly to buy. Dendrites have significant applications in biomedical field because of its characteristics, including hyper-branching, well-defined globular structures, outstanding structural uniformity, multivalency, varying chemical constitution, and higher biological compatibility.

In medical field, mathematical modelling is used to analyse the representation of emerging drugs, normally as an undirected graph, such that each vertex depicts an atom and an edge depicts a link between atoms. Every year new drugs are available and needs remarkable work to select the qualities of the emerging drugs. Dendrimers are a good option in the drug design because of its biological characteristics such as polyvalency, self-assembling, electrostatic interactions, chemical stability, low cytotoxicity, and solubility. The remarkable and emerging role of dendritic macromolecules is in therapies of anticancer and image diagnosis.

Various studies have revealed that there is a consistent correlation between the molecular structures of compounds, drugs, and their characteristics. Topological indices are numerical variants that assist researchers in understanding physical properties, chemical interactions, and biological activity [17–21]. Hence, the discussion on topological indices of chemical structures of drugs helps to know the theoretical basis to prepare new drugs. In this study, SS index is defined and computed for porphyrin (DnPn), propyl ether imine (DPZn), zinc porphyrin (PETIM), and polyethylene amide amine (PETAA) dendrimers [22,23].

In this paper, the notations and terminologies pertaining to the graphs are found in [24].

Definition 1. The oldest and the most studied indices, the first and second Zagreb indices [25], proposed by Gutman and Trinajstić are defined as

Definition 2. Vukičević et al. introduced inverse sum indeg index [26] and stated as

Definition 3. In this work, a novel invariant known as SS (Shilpa-Shanmukha) index is introduced and studied. This index is defined as follows:

Throughout this article, and represent the degrees of vertices and , respectively.

2. Chemical Applicability of the SS Index through QSPR Analysis

Here, we discussed the proposed topological index known as SS index to study the physico-chemical properties, namely, bp, mv, mr, hv, and cp of 67 alkanes ranging from n-butanes to nonanes. The 5 physico-chemical properties of 67 alkane isomers can be found in [27] and Table 1 represents the computed values of four topological indices (SS, ISI, M₁, and M₂) of 67 alkane isomers. The 5 characteristics of alkane isomers are correlated with SS index and it is found that SS index has shown good correlation with all the 5 properties compared to the existing three most popular indices M₁, M₂, and ISI considered in the study. The SS index is plotted against each of the 5 properties of alkane isomers which is depicted in Figure 1.

Table 1: The values of various topological indices of alkanes.

SI. No.
Alkane
SS (G)
ISI (G)
M₁ (G)
M₂ (G)

1
Butane
2.8165
2.6667
10
8

2
2-Methyl propane
2.598
2.25
12
4

3
Pentane
2.9428
3.3333
14
12

4
2-Methyl butane
3.644
3.3667
16
14

5
2,2-Dimethyl propane
3.5777
3.2
20
16

6
Hexane
4.633
4.3333
18
16

7
2-Methyl pentane
4.644
4.3667
20
18

8
3-Methyl pentane
4.69
4.4833
20
19

9
2,2-Methyl butane
4.6545
4.4
24
22

10
2,3-Dimethyl butane
4.6888
4.5
22
21

11
Heptane
5.633
5.3333
22
20

12
2-Methyl hexane
5.644
5.3667
24
22

13
3-Methyl hexane
5.6899
5.4833
24
23

14
3-Ethyl pentane
5.7358
5.6
24
24

15
2,2-Dimethyl pentane
5.6544
5.4
28
26

16
2,3-Dimethyl pentane
5.7347
5.616
26
26

17
2,4-Dimethyl pentane
5.655
5.4
26
24

18
3,3-Dimethyl pentane
5.7312
5.6
28
28

19
Octane
6.633
6.333
26
24

20
2-Methyl heptane
6.644
6.3667
28
26

21
3-Methyl heptane
6.69
6.4833
28
27

22
4-Methyl heptane
6.68
6.49
28
27

23
3-Ethyl hexane
6.736
6.6
28
28

24
2,2-Dimethyl hexane
6.6545
6.4
32
30

25
2,3-Dimethyl hexane
6.7348
6.6166
30
30

26
2,4-Dimethyl hexane
6.7009
6.5166
30
29

27
2,5-Dimethyl hexane
6.655
6.4
32
30

28
3,3-Dimethyl hexane
6.7312
6.6
32
32

29
3,4-Dimethyl hexane
6.7807
6.7333
30
31

30
3-Ethyl-2-methyl pentane
6.7807
6.7333
30
22

31
3-Ethyl-3-methyl pentane
6.808
6.8
32
34

32
2,2,3-Trimethyl pentane
6.7706
6.731
34
35

33
2,2,4-Trimethyl pentane
6.6655
6.4333
34
32

34
2,3,3-Trimethyl pentane
6.8014
6.8143
34
36

35
2,3,4-Trimethyl pentane
6.7796
6.75
32
33

36
Nonane
7.6333
7.3333
30
28

37
2-Methyl octane
7.644
7.3667
32
30

38
3-Methyl octane
7.69
7.4833
32
31

39
4-Methyl octane
7.69
7.4833
32
31

40
3-Ethyl heptane
7.7358
7.6
32
32

41
4-Ethyl heptane
7.7358
7.6
32
32

42
2,2-Dimethyl heptane
7.6545
7.4
36
34

43
2,3-Dimethyl heptane
7.7348
7.6166
34
34

44
2,4-Dimethyl heptane
7.7009
7.5166
34
33

45
2,5-Dimethyl heptane
7.7963
7.7166
34
33

46
2,6-Dimethyl heptane
7.655
7.4
34
32

47
3,3-Dimethyl heptane
7.7312
7.6
36
36

48
3,4-Dimethyl heptane
7.7807
7.7333
34
35

49
3,5-Dimethyl heptane
7.7468
7.6333
34
34

50
4,4-Dimethyl heptane
7.7312
7.6
36
36

51
3-Ethyl-2-methyl hexane
7.7807
7.7333
34
35

52
4-Ethyl-2-methyl hexane
7.9246
8.0333
34
34

53
3-Ethyl-3-methyl hexane
7.808
7.8
36
36

54
2,2,4-Trimethyl hexane
7.7114
7.55
38
37

55
2,2,5-Trimethyl hexane
7.6655
7.4333
38
36

56
2,3,3-Trimethyl hexane
7.8014
7.8143
38
40

57
2,3,4-Trimethyl hexane
7.8255
7.8667
36
44

58
2,3,5-Trimethyl hexane
7.7458
7.65
36
44

59
3,3,4-Trimethyl hexane
7.8845
8
38
41

60
3,3-Diethyl pentane
7.866
7.931
44
40

61
2,2-Dimethyl-3-ethyl pentane
7.878
8.0143
54
40

62
2,3-Dimethyl-3-ethyl pentane
7.8255
7.8666
46
42

63
2,4-Dimethyl-3-ethyl pentane
7.8575
8
44
38

64
2,2,3,3-Tetramethyl pentane
7.8575
8
42
46

65
2,2,3,4-Tetramethyl pentane
7.8154
7.643
40
42

66
2,2,4,4-Tetramethyl pentane
7.676
7.4667
42
40

67
2,3,3,4-Tetramethyl pentane
7.8715
8.0281
40
44

(a)

(b)

(c)

(d)

(e)

Figure 1: Correlation of SS index with properties of alkane isomers.

Regression model for properties of alkane isomers.

The linear regression model is given by

where P is the physical property and TI is the topological index. Equation (4) results in the following linear regression models for various properties with SS index.

The SS index has correlation coefficients of 0.931, 0.98, 0.99, 0.951, and –0.889 with residual standard errors 14.46, 3.76, 0.57, 1.69, and 1.45 and all these models are statistically significant, since the level of significance value of all models is less than 0.05.

Some important observations from the data are presented in Table 2. The correlation coefficients of bp, mv, mr, and hv have shown high positive correlation for the introduced SS index. Also, it is interesting to know that the correlation coefficient for cp shows highly negative correlation for the SS index.

Table 2: The correlation coefficient between SS, ISI, M₁, and M₂ indices with some physico-chemical properties of alkane isomers.

Properties
bp
mv
mr
hv
cp

SS (G)
0.931
0.98
0.99
0.951
−0.889

ISI (G)
0.931
0.96
0.984
0.942
−0.855

M₁ (G)
0.779
0.799
0.837
0.715
−0.684

M₂ (G)
0.816
0.8065
0.854
0.737
−0.688

3. Dendrimers

Dendrimers, come from the Greek word which means “trees,” are branched at the core and they form a spherical three-dimensional structure. Dendrimers have attracted a lot of researchers globally in the study of topological indices [28–33]. The aim of this paper is to compute SS index of four dendrimer structures, namely, DnPn, DPZn, PETIM, and PETAA.

3.1. SS Index of Porphyrin Dendrimer (DnPn)

Consider the porphyrin dendrimer family. This family of dendrimers is denoted by DnPn. The molecular graph of DnPn is shown in Figure 2.

Figure 2: The molecular graph of porphyrin dendrimer.

Let G be the molecular graph of DnPn. By calculation, it is found that G consists of number of vertices and edges to be and , respectively. Table 3 shows the six forms of edges in DnPn(G) based on degrees of end vertices of each edge.

Table 3: Edge partition of DnPn.

, where

Number of edges

Theorem 1. Let DnPn be the family of porphyrin dendrimers. Then, the SS index of DnPn is given by

Proof. From the definition of SS index and Table 3, we deduce

3.2. SS Index of Zinc Porphyrin Dendrimer (DPZn)

Consider the zinc porphyrin dendrimer family. This family of dendrimers is represented by DPZn. The molecular graph of DPZn is depicted in Figure 3.

Figure 3: The molecular graph of zinc porphyrin dendrimer.

Let G be the molecular graph of DPZn. By calculation, it is found that G has vertices and edges. Table 4 shows the four forms of edges in DPZn based on degrees of end vertices of each edge.

Table 4: Edge partition of DPZn.

, where

Number of edges

Theorem 2. Let DPZn be the family of zinc porphyrin dendrimers. Then, the SS index of DPZn is given by

Proof. From the definition of SS index and Table 4, we deduce

3.3. SS Index of Propyl Ether Imine Dendrimer (PETIM)

Consider the family of propyl ether imine dendrimers. This family of dendrimers is represented by PETIM. The molecular graph of PETIM is depicted in Figure 4.

Figure 4: The molecular graph of propyl ether imine dendrimer.

Let G be the molecular graph of PETIM. By calculation, G has vertices and edges. Table 5 shows the three forms of edges in PETIM based on degrees of end vertices of each edge.

Table 5: Edge partition of PETIM.

, where

Number of edges

Theorem 3. Let PETIM be the family of propyl ether imine dendrimers. Then, the SS index of PETIM is given by

Proof. From the definition of SS index and Table 5, we deduce

3.4. SS Index of Polyethylene Amide Amine (PETAA) Dendrimer

Consider the family of polyethylene amide amine dendrimers. This family of dendrimers is represented by PETAA. The molecular graph of PETAA is depicted in Figure 5.

Figure 5: The molecular graph of polyethylene amide amine dendrimer.

Let G be the molecular graph of PETAA. By calculation, G has vertices and edges. Table 6 shows the four forms of edges in PETAA based on degrees of end vertices of each edge.

Table 6: Edge partition of PETAA.

, where

Number of edges

Theorem 4. Let PETAA be the family of zinc porphyrin dendrimers. Then, the SS index of PETAA is given by

Proof. From the definition of SS index and Table 6, we deduce

4. Results and Discussion

In this work, novel topological index known as SS index is introduced and the proposed index is computed for 67 alkane isomers to study the physico-chemical properties, namely, bp, mv, mr, hv, and cp. A linear regression model of these physical properties with SS index is presented. From Table 2 and Figure 1, the SS index has highest correlation with molar refraction (mr) which is . Also, SS index is with boiling point (bp) 0.931, with molar volume (mv) 0.98, with heat of vaporization (hv) 0.951, and with critical pressure (cp) −0.889. From Table 2 by inspection, it is clear that SS index has good correlation with the physico-chemical properties compared to the existing indices, namely, inverse sum indeg index, first Zagreb index, and second Zagreb index. Also, the work focuses on computing the SS index for four dendrimer structures, namely, DnPn, DPZn, PETIM, and PETAA. The values of n are substituted for n = 1 to 10. By inspection from Table 7, it is very clear that SS index increases as n increases. Also, it is observed that correlation coefficient of DnPn is , which is more than the correlation coefficients of DPZn, PETIM, and PETAA which are 0.798837, 0.798841, and 0.798835, respectively. For each of the four structures, a graph as shown in Figure 6 is plotted against the values found in Table 7.

Table 7: Numerical comparison of SS(DnPn), SS(DPZn), SS(PETIM), and SS(PETAA) for n = 1 to 10.

n
1
2
3
4
5
6
7
8
9
10

DnPn
100.60
212.78
324.95
437.13
549.31
661.48
773.66
885.84
998.01
1110.2

DPZn
92.252
231.48
509.95
1066.9
2180.7
4408.4
8863.9
17775
35596
71240

PETIM
23.84
72.251
169.07
362.72
750.02
1524.6
3073.8
6172.2
12369
24762

PETTA
37.69
94.97
209.53
438.65
896.89
1813.4
3646.3
7312.2
14644
29308

Figure 6: Graphical comparison of SS(DnPn), SS(DPZn), SS(PETIM), and SS(PETAA) for n = 1 to 10.

As the SS index is found to have very good correlation coefficient with the above-discussed physico-chemical properties, the novel index is of great use in the QSPR/QSAR/QSTR analysis by the chemists.

5. Mathematical Properties of SS Index

In this section, the SS indexes of cycle, star, path, and simple graphs are computed [34–38].

Theorem 5. For a cycle Cn, where n is the cardinality of vertices, then the SS index of Cn is given by SS(Cn) = n.

Proof. A cycle Cn has n vertices and n edges. The n edges of the cycle will be of type (2, 2). By considering all the n edges and using the definition of SS index, we get SS(Cn) = n.

Theorem 6. For a star Sn, where n is the cardinality of vertices, then the SS index of Sn is given by

Proof. A star Sn has n vertices and (n − 1) edges. The (n − 1) edges of the star graph will be of type (1, n − 1). By considering all the (n − 1) edges and using the definition of SS index, we get

Theorem 7. For a path Pn, where n is the cardinality of vertices, then the SS index of Pn is given by .

Proof. A path Pn has n vertices and (n − 1) edges. The (n − 1) edges of the path graph will be 2 edges of type (1, 2) and (n − 3) edges of type (2, 2), respectively. By considering all the edges and using the definition of SS index, we get .
The SS indexes of cycle, star, and path graphs are related as follows:

Theorem 8. Consider a simple graph G with m edges and cardinality n. Let p, ∆, and δ₁ be the pendent vertices and maximum and minimum vertex degrees of G, respectively. Then,

Proof. For ,

such that the equality holds iff . Also,

with equality holding if . Using Polya–Szego inequality,

we have

For , from (20) and (21), we get

Theorem 9. For a tree T with cardinality n and pendent vertices p, then the SS index is

Theorem 10. Consider simple graph G of order n with m edges and cardinality n. Here, p, ∆, and δ₁ represent pendent vertices and maximum vertex degree and minimum nonpendent vertex degree, respectively, and then

Proof. By Cauchy–Schwarz inequality,

Theorem 11. The cardinality n of a tree T and pendent vertices p and then the SS index is

6. Conclusion

In this article, a novel index known as SS index is introduced and computed for four dendrimers such as DnPn, DPZn, PETIM, and PETAA. To validate the performance of this novel index, the chemical applicability of 67 alkane isomers is studied. It is observed that the proposed index has a very good correlation with alkane isomers considered in the study. The results obtained for the dendrimers have proved that they play a major role in drugs including anti-inflammatory, antimicrobial, and anticancer in administering the drug. Dendrimers play a vital role in the discovery of drugs against the diseases such as Alzheimer’s, HIV, and cancer. The article is concluded by mathematical properties of SS iindexes for cycle, star, path, and simple graphs.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally to this study. M. C. Shanmukha gave the idea and wrote the manuscript. A. Usha and K. C. Shilpa edited and verified the results. Weidong Zhao checked and corrected the initial manuscript and verified the results. M. Reza Farahani added some final remarks and improved the overall paper. All authors read and approved the final draft.

Acknowledgments

This work was supported by the National Key Research and Development Program under Grant 2018YFB0904205. This research received no external funding.

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