Abstract
Nanostar dendrimers are tree-like nanostructures with a well-defined, symmetrical architecture. They are built in a step-by-step, controlled synthesis process, with each layer or generation building on the previous one. Dendrimers are made up of a central core, a series of repeating units or branches, and a surface group shell. A weighted graph is a type of graph in which vertices or edges are assigned weights that represent cost, distance, and a variety of other relative measuring units. The weighted graphs have many applications and properties in a mathematical context. The topological indices are numerical values that represent the symmetry of a molecular structure. They have rich applications in theoretical chemistry. Various topological indices can be used to investigate a wide range of properties of chemical compounds with a molecular structure. They are very important in mathematical chemistry, especially in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies. In this paper, we examine the topological properties of the molecular graphs of nanostar dendrimers. For this purpose, the topological indices, namely, the Wiener index and the Wiener polarity index are computed for a class of nanostar dendrimers.
1. Introduction and Preliminary Results
1.1. Dendrimers
Dendrimers [1, 2] are hyperbranched macromolecules that carry branches from generation to generation with a central part. Dendrimers were first proposed in the late 1970s by German chemist Fritz Vögtle, who proposed highly branched macromolecules with well-defined structures. Vögtle envisioned these synthetic molecules as mimics of natural polymers such as proteins and DNA, with controlled architectures and precise properties. In the early 1980s, American chemist Donald A. Tomalia made significant contributions to the development of dendrimers. Tomalia created the first class of dendrimers, known as polyamidoamine (PAMAM) dendrimers, which is made of repetitively branched subunits of amide and amine functionality on his own. He used a divergent growth method in which he repeatedly reacted on a core molecule with a monomer, resulting in branching and the formation of a well-defined three-dimensional structure.
Dendrimers are iteratively synthesised polymers with unique properties such as polyvalency, electrostatic interaction, self-assembly, monodispersity, stability, and unimolecular micelles that make them an excellent drug delivery carrier. Dendrimer plays an important role in the biomedical field [3] and in gene delivery systems [4].
These nanostar dendrimers were created by fusing two different types of nanoparticles: gold nanostars and dendrimers. The high surface area and unique optical properties of gold nanostars make them ideal for sensing and imaging applications. Dendrimers were chosen because of their ability to transport multiple drug molecules while also targeting specific cells or tissues in the body. To make the nanostar dendrimers, the researchers first synthesised gold nanostars using a seed-mediated growth method. They then functionalized the surface of the gold nanostars with a dendritic molecule called polyamidoamine (PAMAM) dendrimers, which has a large number of functional groups on its surface that can be used to attach drug molecules. The resulting nanostar dendrimers had a core-shell structure, with the gold nanostars forming the core and the dendrimers forming the shell. The researchers demonstrated that these hybrid nanoparticles were effective at delivering drug molecules to cancer cells in vitro and that they could also be used for imaging applications. Dendrimers have received a lot of attention in scientific research and various applications.
1.2. Topological Indices
In cheminformatics, topological indices are mathematical descriptors that are used to measure and characterise molecular structure properties. These indices provide the molecular structure with a numerical depiction by encapsulating details regarding the connectivity and configuration of atoms inside a molecule. Topological indices concentrate on characteristics that are independent of particular spatial arrangements because the name “topological” refers to the study of properties that remain unchanged under continuous deformations. The topological indices are numerical values that represent the symmetry of a molecular structure. In this context, atom distribution and spatial arrangement inside a molecule are referred to as symmetry. It involves the recurrence of motifs or patterns. Because it affects several characteristics, including stability and reactivity, symmetry is a crucial component of molecular structure. They have rich applications in theoretical chemistry. Numerous characteristics of chemical compounds having a molecular structure can be examined using various kinds of topological indices. They play a very crucial role in mathematical chemistry, particularly in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship- (QSPR-) related studies. Many of these topological indices were introduced by researchers in Mathematical Chemistry, on the basis of molecular structure modelling involving graph structures. They sum up some molecular properties in a single numeric value. Topological indices have been used extensively in recent years to study the properties of dendrimers [5–9].
1.3. Wiener and Wiener Polarity Index
According to [2], the Wiener index was the first and is still the most studied topological index. It was chemistry’s first application of a molecular topological index. It is demonstrated that the Wiener index number and the boiling points of alkane molecules are closely connected. Later work on quantitative structure-activity relationships revealed correlations between the critical point’s parameters [10], the density, surface tension, and viscosity of its liquid phase [11], and the molecule’s van der Waals surface area [5].
The Wiener index, indicated mathematically by the symbol , is the sum of all distances between each graph vertex.
Later on, Wiener introduced another descriptor known as the Wiener polarity index that is known to be related to the cluster coefficient of networks. The Wiener polarity index is denoted by and is defined as the number of unordered pairs of vertices that are at distance 3 in . In organic compounds, say paraffin, the Wiener polarity index is the number of pairs of carbon atoms which are separated by three carbon-carbon bonds. Based on the Wiener index and the Wiener polarity index, the formulawas used to calculate the boiling points of the paraffins, where , , and are constants for a given isomeric group. By using the Wiener polarity index, Lukovits and Linert demonstrated quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons in [9]. Hosoya in [11] found a physical-chemical interpretation of . Actually, the Wiener polarity index of many kinds of graphs is studied, such as trees [12], unicyclic and bicyclic graphs [13], hexagonal systems, fullerenes, and polyphenylene chains [6], and lattice networks [14].
1.4. Weighted Graph
The idea of assigning weights to edges and vertices in graphs began to emerge in the early century. Researchers recognized the need to represent relationships with different strengths or costs in various applications. Dénes Ko ig, a Hungarian mathematician, made significant contributions to the study of weighted graphs. In his book “Theory of Graphs and Its Applications” published in 1936, Ko ig discussed concepts such as minimum spanning trees and network flows, which involved the use of weights on edges and vertices.
A weighted graph is a graph together with the weight function . The Wiener index of the weighted graph was introduced in [15] as follows:
For a graph , the Djokovic–Winkler’s relation on [16] is defined as follows:
If , then is related with . Relation is reflexive and symmetric; its transitive closure is an equivalence relation. The partition of induced by will be called the -partition. Assume that denote the -partitions of .
A partition of is coarser than if each set is the union of one or more -classes of [16].
Theorem 1 (see [16]). Let be a connected weighted graph and be partition of coarser than . Then,where ( is defined by for all connected components of .
The technique to construct the quotient graph is to first remove all the edges of from and then shrink the vertices in each component of to one point, where vertex strength is the number of vertices in that component and edge strength is the number of edges in that component. Thus, the number of vertices in is the number of components of and if there exists an edge between the vertices of two components in , then an edge is formed in . In this way, the quotient graph for each partition can be formed.
The following result gives the Wiener index of the graph composed of two subgraphs such that both the subgraphs have one vertex common.
Theorem 2 (see [17]). Let be a graph composed by and graphs such that and = {s}, where is a cut vertex in . Let and . If and , then . And
There are many types of nanostar dendrimers. Dendrimers have received a lot of attention in scientific research and various applications. Dendrimers are regarded as one of the most important, commercially available building blocks in nanotechnology. Dendrimers are used to make nanotubes, nanolatex, chemical sensors, micro- and macrocapsules, coloured glass, modified electrodes, and photon funnels, which are used to make artificial antennas. Because of its widespread application in a variety of fields, researchers have focused their efforts on determining the underlying topology of nanostar dendrimers. The -index of nanostar dendrimers has been calculated by De and Nayeem [18]. In terms of Zagreb indices, Siddiqui et al. [19] investigated the topological properties of some nanostar dendrimers. Bokhary and Tabassum studied different graph invariants to explore different topological properties of dendrimers like the energy of some tree dendrimers [20], domination and power domination of certain dendrimers [21]. The readers are directed to [7, 9, 15, 20–26] for additional discussion in this field. The purpose of this report is to calculate the distance-based topological indices for a class of nanostar dendrimers. The first type of nanostar dendrimer that we study in this work was introduced by Dorosti et al. [25]. In this paper, they computed the Cluj index for two types of dendrimer nanostructures by analyzing the constitutive substructures of these dendrimers. In this paper, we extend this study by computing the Wiener and Wiener polarity indices of the first type of dendrimer mentioned in [25].
2. Main Results
2.1. Construction of the First Type of Nanostar Dendrimer
The stages of the first type of nanostar dendrimers can be made by connecting the multiple hexagons. There are stages of nanostar dendrimer. In the first stage, there are seven connected hexagons which are attached with the nucleus of hexagon. The nucleus of hexagon is made up of five connected hexagons. Thus, at the first stage, there are a total of twelve hexagons. The number of vertices at stage one are and number of edges are .
In the second stage of nanostar dendrimers, eight more hexagons are attached to the first stage. Thus, at the second stage, there are total twenty hexagons. The number of vertices at stage two are and the number of edges are .
The first and second growing stages are different. But, from the third stage and onward, hexagons are attached to the previous stage. Let be the graph of nanostar dendrimer after stages. Thus, for , is obtained from by adding hexagons to . It is easy to see that, for , the order and size of the graph are and , respectively. The graph of nanostar dendrimer of dimension 4 is shown in Figure 1.
2.2. Computation of Wiener Index of Using Vertex Weighted Graph
Let be graph having stages; in the first stage, has one hexagon that is attached to an isolated vertex; after that, is obtained from by adding hexagons, for . For , the order and size of the graph are and , respectively.
It is easy to see that, the graph for has -classes, which can be obtained by applying the Djoković–Winkler relation. The -classes for the graph are shown in Figure 2. Let be the sets coarser than the -classes in . With the help of these sets, the quotient graphs , for , are computed. The quotient graphs and are shown in Figures 3 and 4, respectively.
In the next theorem, the Wiener index of the graph is computed.
Theorem 3. Let be the positive integer, then
Proof. Let be the partition of coarser than . Then, from Theorem 1,where is defined by for all connected components of . This implies that, the weights of vertices and are and 1, respectively, and are computed by using the quotient graph . For , and in quotient graph are and , respectively.
Now, by using equation (2), we haveThus,For and using equation (2) and Theorem 1, we have .Thus, we havewhereReplacing these values in equation (10), we obtainAn easy simplification impliesBy taking summation on , we obtainIt is easy to compute thatBy putting these values in equation (14), we haveHence,By adding equations (8) and (17) and using Theorem 1, we obtainThis implies thatThis completes the proof.
For , the graph has five components. One component is and let the other components be , and such that , for . Thus, is obtained by the edge disjoint copies of and the graph . This implies that .
The partition of the graph is shown in Figure 5.
Define, = , = , = , and = , where , , , and .
It is important to note that , , and .
Further, suppose that = , , = , = , = , and = .
It is easy to compute that and .
In the following lemmas, is computed.
Lemma 4. Let be the positive integer and ,
Proof. The construction of the graph shows that the graph has stages, and in each stage, there are hexagons, for . The hexagons at each stage are connected to two hexagons in the next stage through a vertex. Let be a vertex connecting hexagon in stage to the hexagon at stage. The distance of to this vertex in stage is , where . Further, the distance of to the six vertices of hexagon is . This implies that
Lemma 5. Let be the positive integer, then
Proof. Since , therefore, we haveHence,
Lemma 6. Let be the positive integer, then
Proof. Since , therefore, we haveHence, we havethis completes the proof.
Lemma 6. Let be the positive integer, then
Proof. Since , therefore, we haveHence, we haveThis completes the proof.
An easy computation implies that
Lemma 7. The Wiener index of the graph is .
Now, we prove the main result of this section.
Theorem 8. For , .
Proof. Since , , , and , therefore, by using Theorem 2, we obtainBy replacing the values of , and in recursively, we obtainSince , therefore,By putting values of and and and from Lemmas 5, 6, and 7 in the above equation, we obtainSinceTherefore, Wiener index of reduces to .
The last expression after simplification becomesBy replacing this value, Wiener index of reduces toHence, . This completes the proof.
2.3. Wiener Polarity Index of
From the construction of the graph , it is easy to see that
Theorem 9. For ,
Proof. For , the pair of vertices at distance 3 in the hexagon is 3, and there is no vertex at distance three between two hexagons at the same stage. The pair of vertices at distance 3 between the hexagons of stages and is 18, and there is no pair of vertices at distance 3 from stages to . Thus,
3. Concluding Remarks
In this article, the Wiener index and Wiener polarity index of a class of nanostar dendrimers are explored. We have computed the exact values of the Wiener index and the Wiener polarity index of the nanostar dendrimer . Figure 6 shows the graphical comparison of both indices. These topological indices provide a valuable tool to measure dendrimer molecular structure, allowing for comparisons, forecasts, and comprehension of the potential effects of various structural aspects on behaviour and qualities. The solubility, stability, and reactivity of dendrimers can be shown by particular topological indices. By establishing correlations, properties dependent on the topological structure of the dendrimer can be predicted. Relationships between structure and property can be established with the help of topological indices. In future, researchers can determine which features of the dendrimer’s structure have the most impact on how it behaves or performs.
Data Availability
Data from previous studies were used to support this study.
Disclosure
This work is part of the MS dissertation of one of the co-authors, Pakizah Bashir [22].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Larg Groups Project under grant number RGP.2/395/44. This study was supported via funding from Prince Sattam bin Abdulaziz University project number PSAU/2023/R/1444.