Abstract
Dendrimers are well-defined nanoparticles, which have far-reaching application in the field of chemistry. Many efforts have been devoted to development of dendrimer due to thier unique structure and various properties and broad application. It helps in varieties of purposes as a catalyst in drug delivery and drug design. The topological descriptor analyze the structure–property relationship of chemical compounds. In this paper, some numerical expressions have been obtained to understand the behavioral pattern of chiral polyamidoamine (PAMAM) dendrimer and PAMAM anthracene moieties dendrimer. The analytical expression has been plotted and compared with varieties of indices to show how it varies between each indices.
1. Introduction
The topological descriptor helps to analyze the chemical structure and their physical properties. It is essential to understand the operational efficiency, performance, and side effects of new drugs in order to track the process and improve the impact. This helps the medicinal chemistry to overcome the insufficient funds and reach their target at reduced cost. New techniques could help to overcome these challenges.
Nanoparticles have been in use for decades in various fields. The word “nano” refers to their size of 1–100 nm. They are capable of carrying drugs and biomolecules and are used in nanotherapy. Vogtle and his coworkers introduced a peculiar nanostructure, dendrimer, in the year 1978. Tomalia et al. [1], Newkome et al. [2], and numerous research group contributed to the development of a new dendrimer. Dendrimers have been extensively investigated and are currently applied in various fields of drug delivery [3, 4], catalysis [5], light harvesting, and porous material [6]. Dendrimers are composite structure, but are well-defined multibranched chemical composition, with high degree of order containing selected chemical units in predetermined sites of their tree-like structure [7]. Dendrimers have a branched architecture and three-dimensional spatial arrangement, and their extension is still predictable. From a topological standpoint, dendrimers generally consist of three structural phases: single central core, surrounded by bridges called generations composed of repeating units and terminated by functional/peripheral groups. Dendrimers are synthesized using a stepwise repetitive reaction sequence, which happens in two approaches: divergent synthesis and convergent synthesis. In divergent synthesis, the dendrimers are prepared from the core as starting point and buildup the generation. By using the same approach, the first dendrimer polyamidoamines (PAMAMs) had been synthesized. Convergent synthesis starts from the surface and ends up at the core. This approach does not allow to form the higher generations because steric problems occur in reaction. With the emergence of a large number of applications in anticancer, anti-inflammatory, antimicrobial activities there is a critical need to study and understand its properties. For application of dendrimer and further details, see [8–12].
Topological indices are numerical parameters that determine the physicochemical properties of compounds. Harry Wiener introduced the method called Wiener index in the year 1947 [13]. In evolving QSAR/QSPRs, the structural properties of chemical compounds such as photophysical, pharmacological, and pharmacokinetic are obtained by topological indices. The chemical properties such as molecular refraction, critical temperature, boiling point, critical pressure, partition coefficients, osmotic coefficients, and viscosity can be acquired using topological indices [14, 15]. Therefore, each topological indices help to analyze and quantify different physical compositions of a chemical structure and also the chemical networks [16].
In graph theory, the molecular structure of drug can be represented as graph. In chemical drug compounds, how the topological indices are applied and used were explained by García et al. [15]. Dendrimers are demanded in the industry of drug delivery and design due to its properties such as hyperbranching, photosensitizer, targeted delivery, multivalency, structure uniformity, solubility, and low toxicity and also used as anticancer and antitumor therapies.
Chiral dendrimers have also been prepared by employing synthesis of monomers. To date, the chiral dendrimer can be categorized into different classes: dendrimer possessing a chiral core and achiral branches; dendrimer possesing chiral peripheral surface group; dendrimer constructed from an achiral core and constitutionally different branches; dendrimer featuring chiral branched units; and dendrimers featuring chiral core, asymmetric branching units, and optically active chiral core. In this study, the dendrimers possesing chiral peripheral surface group: chiral PAMAM dendrimer and PAMAM anthracene moieties dendrimer have been considered.
Chiral PAMAM dendrimer is a nanocatalysis tetracarboxylic acid core with four branches, as shown in Figure 1 [17]. PAMAM anthrancene moieties are a blue light-emitting ethylenediamine core with four branches, as shown in Figure 2 [18]. Both were peripherally substituted PAMAM dendrimers. The surface of the PAMAM structure is achiral and hydrophobic in nature and it has both light-emitting and electrodeposition properties. The generation of chiral PAMAM dendrimer and PAMAM anthracene moieties dendrimer has been denoted as and , respectively. By shrinking the original graph into quotient graph, the distance-based topological indices of and can be determined.
2. Topological Descriptor
In this paper, let us consider connected graphs. For basic terminologies, refer to Bondy and Murty [19]. For an edge , and be the set of vertices and edges near to the vertex than and their cardinalities are denoted as and , respectively. The strength-weighted graph concept is introduced by Arockiaraj et al. [20], where is the vertex weight, is the vertex strength, and is the edge strength. In the strength-weighted graph, we define and and the cardinality of and , where . Let . We represent as .
Some preliminary concepts are defined, which are the key tool in finding the topological indices. Let us consider a graph , the Djoković–Winkler’s relation on , [21, 22] is as follows. If , then is related with . The relation is always reflexive and symmetric and its transitive closure is an equivalence relation. Using partition, we convert the original graph into the quotient graph as given by Klavžar et al. [23]. A partition of is coarser than = , if is the union of one or more sets in . We refer to [13, 21, 24–27] for the basic definitions of distance-based topological indices. For the distance-based topological indices of strength-weighted graph, refer to Table 1.
Theorem 1 (see [27]). For a connected strength-weighted graph , let be a partition of coarser than . Let and . Then,where(i) is defined by (ii) is defined by (iii) is defined as the number of edges in such that one end in and the other end in , for any two connected components C and D of -rooted product of graphsand, denoted by, is the graph obtained by joining the root nodes ofcopies of to each vertex of.
Theorem 2 (see [28]). Let be a connected weighted graph, and let be the root node of , where , . Then,where .
Chiral PAMAM dendrimer, denoted by , has vertices and edges. consists of classes. Figure 3 represents the demonstration of chiral PAMAM dendrimer of generation. We have the sets coarser than the classes. By shrinking the original graphs to quotient graphs, four types of quotient graphs are obtained. The four types are demonstrated in Figures 4–7.
The first quotient graph, , has five vertices and four edges. In that, three vertices have vertex weight 1 and vertex strength 0, where other two vertices have vertex weight and vertex strength , respectively, as shown in Figure 4.
The second quotient graph, , where , contains vertices and edges. It has branches that are connected to with . The vertex weight and vertex strength of the vertices of all branches are 1 and 0, except vertices ; with , respectively, as shown in Figure 5.
The third quotient graph, , contains vertices and edges. It has branches which are connected to with . Except the vertices ; with , the vertex weight and vertex strength of all the vertices of branches are 1 and 0, as shown in Figure 6.
The fourth quotient graph, , contains vertices and edges. It has hexagons that are merged to with . The vertex weight and vertex strength of all the vertices are 1 and 0, respectively, as shown in Figure 7.
Theorem 3. If , then .
Proof. The Wiener index of each quotient graph is calculated as follows:We use the Theorem 2 to compute the Wiener index of the following quotient graphs. For and where .Therefore,Also,where .where .Therefore,In the next theorem, we compute Szeged-related indices.
Theorem 4. If , then .
Proof. The vertex Szeged index of is calculated by evaluating vertex Szeged index of each quotient graph.For ,where .where .where .Therefore,The edge Szeged index of is calculated by evaluating edge Szeged index of each quotient graph.For ,where .where .where .Therefore,The edge-vertex Szeged index of is calculated by evaluating edge-vertex Szeged index of each quotient graph.where .For ,where where .where .Therefore,The total Szeged index of is calculated by evaluating total Szeged index of each quotient graph.Therefore,In the next theorem, we compute Mostar-related indices.
Theorem 5. If , then .
Proof. The Mostar index of is calculated by evaluating Mostar index of each quotient graph.For ,where .where .where .Therefore,The edge Mostar index of is calculated by evaluating edge Mostar index of each quotient graph.For ,where .where .where .Therefore,The total Mostar index of is calculated by evaluating total Mostar index of each quotient graph.Therefore,
Theorem 6. If , then .
Proof. The Padmakar–Ivan index of is calculated by evaluating edge Padmakar–Ivan index of each quotient graph.For ,where .where .where .Therefore,
PAMAM anthracene moieties dendrimer, denoted by , has vertices and edges. consists of classes. Figure 8 represents the demonstration of PAMAM anthracene moieties dendrimer of generation. We have three sets coarser than the classes. By shrinking the original graph to quotient graph, three types of quotient graphs are obtained. The three types are indicated in Figures 9–11.
The first quotient graph, , has four vertices and three edges. In that, two vertices have vertex weight 1 and vertex strength 0, where other two vertices have vertex weight and vertex strength , respectively, as shown in Figure 9.
The second quotient graph, , where , contains vertices and edges. It has branches that are connected to with . The vertex weight and vertex strength of the vertices of all branches are 1 and 0, except vertices ; , with , respectively, as shown in Figure 10.
The third quotient graph, , contains vertices and edges. It has hexagon that is merged to with . The vertex weight and vertex strength of all the vertices are 1 and 0, respectively, as shown in Figure 11.
Theorem 7. If , then
Proof. The proof is similar to Theorem 3.
Theorem 8. If , then .
Proof. The proof is similar to Theorem 4.
Theorem 9. If , then .
Proof. The proof is similar to Theorem 5.
Theorem 10. If , then .
Proof. The proof is similar to Theorem 6.
3. Discussion
In this section, the numerical results for all the indices of whose analytical expressions are determined. Refer to Tables 2–5 for the same. The graphical comparison between Wiener, Padmakar–Ivan, Szeged, and Mostar indices helps to understand the molecular structure of and and its properties. Refer to Figures 12–15 for the same. The graphical behavior of topological indices has been plotted using MATLAB.
4. Conclusion
The distance-based topological indices have been analyzed for and , which help to determine its structural characterization and properties. Dendrimers are emerging in diverse areas such as material science and biomedical science. The numerical results of and help to integrate the indices, which ease to examine the synthesis of dendrimer.
Data Availability
All the data and material used in this research are included in the paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.