Wiener Polarity Index Calculation of Square-Free Graphs and Its Implementation to Certain Complex Materials

School of Library, Anhui Jianzhu University, Hefei 230601, China College of Economics, Sichuan Agricultural University, Chengdu 610000, China Department of Mathematics, Loyola College, Chennai 600034, India Department of Mathematics, Loyola College, University of Madras, Chennai 600034, India Department of Mathematics, Sri Venkateswara College of Engineering, Sriperumbudur, Kanchipuram 602117, India


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][2][3]. e 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. e 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. e 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][7][8][9][10]. e 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. e 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 G � (V(G), E(G)) be a connected graph. e cardinality of the vertex set and the edge set of G is represented by |V(G)| and |E(G)|, respectively. e number of pentagons and hexagons of G is denoted by N p (G) and N h (G), respectively. For any positive integer i, we represent N i |. e first and second Zagreb indices of a graph G are defined, respectively, as follows: (1) e Wiener polarity index of a graph G is defined as It was observed that W P (G) � (1/2) v∈V(G) |N 3 G (v)| [11], and the importance of W P has been demonstrated in various papers [9,10,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In this paper, we study the Wiener polarity index of C 4 -free graphs and implement our technique to cancer treatment drugs, silicate, Sierpiński, and octahedral-related networks.

Derivation of the Key Result
For a tree T, it was realized [18] that , and interms of Zagreb indices can be rewritten as For a graph G, where G is C 3 -free and C 4 -free such that its different cycles have at most one common edge [14], . Suppose G is the C 3 -free graph such that its different cycles have at most one common edge [29], where N q i (G) is the number of quadrangles of type i such that the sum of degrees on the vertices of that type quadrangle is K i . In this section, we fill a gap by computing Wiener polarity indices of C 4 -free graphs. Theorem 1. Let G be a C 4 -free graph such that its different cycles have at most one common edge. For uv ∈ E(G), let Proof. Suppose that, for any edge uv of G that k uv � 0, it is easy to see that [14,29] Assume that there exists an edge uv ∈ E(G) such that k uv ≠ 0, as shown in Figure 1. en, the number of pairs of vertices x, y of G such that d G (x, y) � 3 and containing the edge uv as the internal edge is

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.
Proof. Since lenvatinib and cabozantinib are square-free graphs, we complete the proof by using eorem 1 and Table 1.
Mathematical Problems in Engineering � 48.
x p y q Figure 1: Edge uv of G such that k uv ≠ 0.

Mathematical Problems in Engineering
In a similar way, we can complete the proof of cabozantinib.

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. e silicate-related [12,31,32] and Imran Sabeel-E-Hafi [33][34][35] networks are depicted in  e removal of silicon vertices (solid) from the silicaterelated networks resulting with oxide type networks are shown in Figures 6-8.
Recently, W P of oxide OX(n) and silicate SL(n) frameworks of dimension n have been computed in [12] using the third neighborhood of vertices as W p (OX(n)) � 63n 2 − 87n − 3 and W p (SL(n)) � 153n 2 − 99n − 3. Now, one can easily obtain these two results by putting k � 1 and 2, respectively, in Corollary 1. We use DSL(n), RTSL(n), DOX(n), and RTOX(n) to represent the n-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.

Octahedral Structures.
e 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 Mathematical Problems in Engineering atom, characterizing the vertices of an octahedron. e 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 SF 6 and molybdenum hexacarbonyl Mo(CO) 6 . e 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. e 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]. e octahedral network and its dominated version are depicted, respectively, in Figures 11 and 12, whereas the rectangular form is portrayed in Figure 13.

Theorem 5.
e Wiener polarity indices of octahedral OT(n) and dominating octahedral DOT(n) networks are given by the following: Proof.

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 e figures, tables, and other data used to support the findings of this study are included within the article.