Abstract
Molecular topology is a portion of mathematical chemistry managing the logarithmic portrayal of chemical materials, permitting a tremendous yet straightforward characterization of the compounds. Concerning the traditional physical-chemical descriptors, it is conceivable to set up direct quantitative structure-activity relationship methods to associate with such descriptors termed topological indices. In this study, we have developed the mathematical technique to study the Wiener polarity index of chemical materials without squares. We have taken the cancer treatment drugs such as lenvatinib and cabozantinib to illustrate our approach. In addition, we explored the inherent property of silicate, Sierpiński, and octahedral-related complex materials that the edge set can be decomposed in such a way that any edge in the same part of the decomposition has an equal number of neighboring vertices and applied the technique to derive the formulae for these materials.
1. Introduction
Drug discovery is the procedure through which potential new helpful substances are established by utilizing a combination of computational, experimental, translational, and clinical models. Regardless of advances in biotechnology and comprehension of organic frameworks, drug discovery is as yet an extensive, costly, complicated, and inefficient process with a high attrition rate of new therapeutic discovery. Drug design is the inventive process of finding new drugs depending on the information of a biological target. Currently, there are several incredible methodologies for drug design and drug database screening [1–3]. The quantitative structure activity and property relationships (QSAR/QSPR) are mathematical models that endeavor to transmit the structure-derived features of a compound to its biological or physico-chemical movement. The relevance of QSAR/QSPR advance in scientific research starts with the idea of the assortment of the input data from databases, then the explanation of molecular descriptors which describes important information of the composed molecules, and through an explicit method to decrease the number of descriptors to the most instructive descriptors, and the last part is the chemoinformatic tools which are used to assemble models that describe the pragmatic relationship between the structure and property or activity.
A significant development in the automated computer treatment of chemical materials and QSAR has been the application of a numerical procedure, namely, graph theory to chemistry. In chemical graph theory, molecular structures are represented as hydrogen-suppressed graphs, regularly known as molecular graphs, in which the atoms are taken by vertices and the bonds by edges. The associations between molecules can be portrayed by different kinds of topological matrices, for example, distance or adjacency matrices, which can be scientifically converted into real numbers and called topological indices. Topological indices (TIs) are usually considered the atomic arrangement of compounds such as atomic size, shape, branching, nearness of heteroatoms, and various bonds [6–10]. The handiness of TIs in QSPR and QSAR contemplates has been broadly illustrated [4], and they likewise have been utilized as a proportion of auxiliary structural similarity or diversity by their application to databases created by PC. The idea of topological indices originated from the work of Wiener, while he was functioning on the paraffin boiling points using Wiener and Wiener polarity indices [5].
All the graphs argued in this paper are simple and finite. Let be a connected graph. The cardinality of the vertex set and the edge set of is represented by and , respectively. The number of pentagons and hexagons of is denoted by and , respectively. For any positive integer , we represent as the neighborhood of , and thus, clearly the open neighborhood of (denoted by ) is , and the degree of (denoted by ) is . The first and second Zagreb indices of a graph are defined, respectively, as follows:
The Wiener polarity index of a graph is defined as
It was observed that [11], and the importance of has been demonstrated in various papers [9, 10, 12–28]. In this paper, we study the Wiener polarity index of -free graphs and implement our technique to cancer treatment drugs, silicate, Sierpiński, and octahedral-related networks.
2. Derivation of the Key Result
For a tree , it was realized [18] that , and interms of Zagreb indices can be rewritten as . For a graph , where is -free and -free such that its different cycles have at most one common edge [14], . Suppose is the -free graph such that its different cycles have at most one common edge [29], , where is the number of quadrangles of type such that the sum of degrees on the vertices of that type quadrangle is . In this section, we fill a gap by computing Wiener polarity indices of -free graphs.
Theorem 1. Let be a -free graph such that its different cycles have at most one common edge. For , let . Then,
Proof. Suppose that, for any edge of that , it is easy to see that [14, 29] . Assume that there exists an edge such that , as shown in Figure 1. Then, the number of pairs of vertices of such that and containing the edge as the internal edge is . Since and , we have . Hence,
Corollary 1. Let be a -free graph such that its different cycles have at most one common edge. If every edge such that , then
3. Numerical Computation
In this section, we compute the numerical quantities for Wiener polarity indices of lenvatinib and cabozantinib cancer drugs and chemical materials based on the silicate, Sierpiński, and octahedral frameworks.
3.1. Cancer Treatment Drugs
Lenvatinib and cabozantinib [30] are orally accessible multikinase inhibitor and antineoplastic specialist that are utilized in the treatment of advanced, metastatic medullary thyroid cancer (MTC) and hepatocellular carcinoma (HCC). Lenvatinib is a member of the class of quinolines, that is, the carboxamide of 4-3-chloro-4-[(cyclopropylcarbamoyl)amino]phenoxy-7-methoxyquinoline-6-carboxylic acid. Cabozantinib is a dicarboxylic acid diamide, that is, N-phenyl-N’-(4-fluorophenyl)cyclopropane-1,1-dicarboxamide in which the hydrogen at position 4 on the phenyl ring is substituted by an (6,7-dimethoxyquinolin-4-yl)oxy group. The molecular graph structures of lenvatinib and cabozantinib are shown in Figure 2.
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Theorem 2. Let and be the graphs of lenvatinib and cabozantinib. Then, , and .
Proof. Since lenvatinib and cabozantinib are square-free graphs, we complete the proof by using Theorem 1 and Table 1. Let such that :In a similar way, we can complete the proof of cabozantinib.
3.2. Silicate and Sierpiński Graphs
Inorganic networks based on silicates and fractal types, three-dimensional metal-catecholate frameworks, metal-organic frameworks, and reticular chemistry as a whole are emerging as cutting-edge fields of research in catalysis and ultrahigh proton conductivity. The silicate-related [12, 31, 32] and Imran Sabeel-E-Hafi [33–35] networks are depicted in Figures 3–5.
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The removal of silicon vertices (solid) from the silicate-related networks resulting with oxide type networks are shown in Figures 6–8.
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Recently, of oxide and silicate frameworks of dimension have been computed in [12] using the third neighborhood of vertices as and . Now, one can easily obtain these two results by putting and 2, respectively, in Corollary 1. We use , , , and to represent the -dimensional dominating silicate, regular triangulene silicate, dominating oxide, and regular triangulene oxide networks, respectively. We now recall the Zagreb indices of silicate- and oxide-related structures in Table 2.
We noted that the Zagreb indices of the dominating silicate network were computed with errors in [31] and can be readily corrected from Table 3.
Theorem 3. The Wiener polarity indices of silicate- and oxide-related structures are given by the following:(1)(2)(3)(4)
Proof. Since silicate- and oxide-related structures are pentagons-free chemical graphs, we have . Moreover, the end vertices of any edge in and have exactly two common neighbors, whereas and have one common neighbor. We can easily see that the number of hexagons in and are equal and the same truth for and structures i.e., and . Based on the above values and by Table 2, we compute the Wiener polarity indices as follows:
We now give the brief definitions of Sierpiński and its gasket graphs and followed by computations of . The 1-dimensional Sierpiński graph is a complete graph on three vertices, and an -dimensional Sierpiński graph is constructed by connecting three copies of with three bringing edges, as shown in Figure 9.
The Sierpiński gasket is obtained from by contracting all its bridging edges (Figure 10). The first and second Zagreb indices of and were computed in [37], and -Zagreb index was dealt in [38]. The number of vertices and edges of are and , respectively. Similarly, comprises of vertices and edges.
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Theorem 4. For , the Wiener polarity indices of Sierpiński graph and the Sierpiński gasket are given by the following:(1)(2)
Proof. We complete the proof by using Theorem 1 and Table 4. Now, we derive the first formula, and the second one can be obtained similarly. Let :
3.3. Octahedral Structures
The idea of octahedral coordination geometry was created by Alfred Werner in 1913 for which he was awarded the Nobel Prize in Chemistry [39]. He clarified the stoichiometries and isomerism in the coordination mix using octahedral coordination geometry. His understanding permitted scientists to legitimize the number of isomers of coordination mixes. Octahedral progress metal buildings containing amines and basic anions are regularly alluded to as Werner-type edifices.
In chemistry, octahedral molecular geometry portrays the shape of compounds with six atoms or gatherings of atoms or ligands symmetrically arranged around a focal atom, characterizing the vertices of an octahedron. The octahedron is one of the platonic solids, even though octahedral molecules commonly have an atom in their centre and no bonds between the ligand atoms. A perfect octahedron fits the point group OH. Illustrations of octahedral compounds are sulfur hexafluoride and molybdenum hexacarbonyl Mo(CO)6. The term octahedral is used somewhat lightly by chemists, concentrating on the geometry of the bonds to the central atom and not considering modifications between the ligands themselves.
The octahedral network was very recently introduced in [40]. Here, we extend the network to its rectangular form with the help of the idea adopted in [41]. The octahedral network and its dominated version are depicted, respectively, in Figures 11 and 12, whereas the rectangular form is portrayed in Figure 13.
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Theorem 5. The Wiener polarity indices of octahedral and dominating octahedral networks are given by the following:(1)(2)
Proof. The Zagreb indices of have been obtained [40] as and . Using Table 5, we can easily derive the Zagreb indices of as and . Moreover, the end vertices of any edge in and have exactly two common neighbors, and we have
Theorem 6. The Zagreb and Wiener polarity indices of rectangular octahedral networks are given by the following:(1)(2)(3)
Proof. For , even and odd, we have where , , and . In this case, the number of hexagon in is , and the end vertices of any edge has exactly two common neighbors:For all other values of and in , we have , , , and . Then,
We have shown the graphical plots of our computed results in Figures 14 and 15.
4. Conclusion
In this paper, we have derived the technique to find the Wiener polarity indices of graphs without squares, and consequently, we have computed the Wiener polarity indices of chemical structures of lenvatinib and cabozantinib, which are used in the treatment of thyroid cancer and HCC. As measured topological indices are proficient at forecasting different properties and behaviors such as boiling point, entropy, enthalpy, and critical pressure, our results can be useful in designing new drugs and vaccines for cancer. In addition to this, we have computed the Wiener polarity indices of some special classes of graphs, namely, silicate, Sierpiński, and octahedral structures with the help of our extended result.
Data Availability
The figures, tables, and other data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.