Computing SS Index of Certain Dendrimers

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.


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.
ese 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 lowincome countries, there is an insufficient funding to cover the costs of reagents and equipment needed to calculate biochemical properties. e 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. e number of vertices and edges of a chemical compound counts to the computation of topological indices [1][2][3][4][5][6][7][8].
A topological index is a computational parameter derived from the graph structure mathematically [9][10][11][12][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. e 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. e indices are of great use in the study of pharmacology, toxicology, and chemistry (QSAR/QSPR/ QSTR) [14][15][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. e 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. ese thoroughly tailored architectural nanomolecules can be functionalized and modify their physico-chemical or biological characteristics. e 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. e 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][18][19][20][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 (D n P n ), propyl ether imine (DPZ n ), 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.
e oldest and the most studied indices, the first and second Zagreb indices [25], proposed by Gutman and Trinajstić are defined as (1) [26] and stated as Definition 3. In this work, a novel invariant known as SS (Shilpa-Shanmukha) index is introduced and studied. is index is defined as follows: roughout this article, d v and d ω represent the degrees of vertices v and ω, respectively.

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. e 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, M 1 , and M 2 ) of 67 alkane isomers. e 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 M 1 , M 2 , and ISI considered in the study. e SS index is plotted against each of the 5 properties of alkane isomers which is depicted in Figure 1.
Regression model for properties of alkane isomers. e 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.     Table 2. e 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.

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][29][30][31][32][33]. e aim of this paper is to compute SS index of four dendrimer structures, namely, D n P n , DPZ n , PETIM, and PETAA.

SS Index of Porphyrin Dendrimer (D n P n ).
Consider the porphyrin dendrimer family. is family of dendrimers is denoted by D n P n . e molecular graph of D n P n is shown in Figure 2.
Let G be the molecular graph of D n P n . By calculation, it is found that G consists of number of vertices and edges to be 96n − 10 and 105n − 11, respectively. Table 3 shows the six forms of edges in D n P n (G) based on degrees of end vertices of each edge.

Theorem 1. Let D n P n be the family of porphyrin dendrimers. en, the SS index of D n P n is given by
Proof. From the definition of SS index and Table 3, we deduce  □

SS Index of Zinc Porphyrin Dendrimer (DPZ n ).
Consider the zinc porphyrin dendrimer family. is family of dendrimers is represented by DPZ n . e molecular graph of DPZ n is depicted in Figure 3.
Let G be the molecular graph of DPZ n . By calculation, it is found that G has 56 × 2 n − 7 vertices and 64 × 2 n − 4 edges. Table 4 shows the four forms of edges in DPZ n based on degrees of end vertices of each edge.
Proof. From the definition of SS index and Table 4, we deduce

SS Index of Propyl Ether Imine Dendrimer (PETIM).
Consider the family of propyl ether imine dendrimers. is family of dendrimers is represented by PETIM. e molecular graph of PETIM is depicted in Figure 4. Let G be the molecular graph of PETIM. By calculation, G has 24 × 2 n − 23 vertices and 24 × 2 n − 24 edges. Table 5 shows the three forms of edges in PETIM based on degrees of end vertices of each edge.
Proof. From the definition of SS index and Table 5, we deduce □

SS Index of Polyethylene Amide Amine (PETAA)
Dendrimer. Consider the family of polyethylene amide amine dendrimers. is family of dendrimers is represented by PETAA. e molecular graph of PETAA is depicted in Figure 5. Let G be the molecular graph of PETAA. By calculation, G has 44 × 2 n − 18 vertices and 44 × 2 n − 19 edges. Table 6 shows the four forms of edges in PETAA based on degrees of end vertices of each edge.
Proof. From the definition of SS index and Table 6, we deduce

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 0.99. 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, D n P n , DPZ n , PETIM, and PETAA. e 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 D n P n is r � 1, which is more than the correlation coefficients of DPZ n , 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.
As the SS index is found to have very good correlation coefficient r � 0.99 � 1 with the above-discussed physicochemical properties, the novel index is of great use in the QSPR/QSAR/QSTR analysis by the chemists.
Theorem 5. For a cycle C n , where n is the cardinality of vertices, then the SS index of C n is given by SS(C n ) � n. Proof. A cycle C n has n vertices and n edges. e 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(C n ) � n. □ Theorem 6. For a star S n , where n is the cardinality of vertices, then the SS index of S n is given by Proof. A star S n has n vertices and (n − 1) edges. e (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 □ Figure 5: e molecular graph of polyethylene amide amine dendrimer.
Proof. A path P n has n vertices and (n − 1) edges. e (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 (n − 1) edges and using the definition of SS index, we get SS(P n ) � (n − 3) ). e SS indexes of cycle, star, and path graphs are related as follows: Proof.
such that the equality holds iff d ] � d ω � Δ. Also, For Δ ≥ d ] , 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 δ 1 represent pendent vertices and maximum vertex degree and minimum nonpendent vertex degree, respectively, and then