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

Authors: Lin Zhang; Tian-Le Sun; Micheal Arockiaraj; M. Arulperumjothi; S. Prabhu
Journal: Mathematical Problems in Engineering
 (2021)

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## 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.


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## Body

## 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. LetG=VG,EG be a connected graph. The cardinality of the vertex set and the edge set of G is represented by VG and EG, respectively. The number of pentagons and hexagons of G is denoted by NpG and NhG, respectively. For any positive integer i, we represent NGiv=u∈VG:dGu,v=i as the ith neighborhood of v, and thus, clearly the open neighborhood of v (denoted by NGv) is NG1v, and the degree of v (denoted by dv) is NG1v. The first and second Zagreb indices of a graph G are defined, respectively, as follows:(1)M1G=∑uv∈EGdu+dv=∑v∈VGdv2,M2G=∑uv∈EGdu×dv.The Wiener polarity index of a graphG is defined as(2)WPG=u,v:dGu,v=3,u,v∈VG.It was observed thatWPG=1/2∑v∈VGNG3v [11], and the importance of WP has been demonstrated in various papers [9, 10, 12–28]. In this paper, we study the Wiener polarity index of C4-free graphs and implement our technique to cancer treatment drugs, silicate, Sierpiński, and octahedral-related networks.


## 2. Derivation of the Key Result

For a treeT, it was realized [18] that WPT=∑uv∈ETdu−1×dv−1, and interms of Zagreb indices can be rewritten as WPT=M2T−M1T+ET. For a graph G, where G is C3-free and C4-free such that its different cycles have at most one common edge [14], WPG=M2G−M1G+EG−5NPG−3NhG. Suppose G is the C3-free graph such that its different cycles have at most one common edge [29], WPG=M2G−M1G+EG−5NpG−3NhG−∑iKi−4NqiG, where NqiG is the number of quadrangles of type i such that the sum of degrees on the vertices of that type quadrangle is Ki. In this section, we fill a gap by computing Wiener polarity indices of C4-free graphs.Theorem 1.
LetG be a C4-free graph such that its different cycles have at most one common edge. For uv∈EG, let kuv=NGu∩NGv. Then,(3)WPG=∑uv∈EGdu−kuv−1×dv−kuv−1−5NpG−3NhG.Proof.
Suppose that, for any edgeuv of G that kuv=0, it is easy to see that [14, 29] WPG=∑uv∈EGdu−1×dv−1−5NPG−3NhG. Assume that there exists an edge uv∈EG such that kuv≠0, as shown in Figure 1. Then, the number of pairs of vertices x,y of G such that dGx,y=3 and containing the edge uv as the internal edge is xi,yj:1≤i≤p,1≤j≤q. Since du=p+kuv+1 and dv=q+kuv+1, we have pq=du−kuv−1dv−kuv−1. Hence,(4)WPG=∑uv∈EGdu−kuv−1dv−kuv−1−5NpG−3NhG.Figure 1
Edgeuv of G such that kuv≠0.Corollary 1.
LetG be a C4-free graph such that its different cycles have at most one common edge. If every edge uv∈EG such that kuv=k≥0, then(5)WPG=M2G−k+1M1G+k+12EG−5NpG−3NhG.


## 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.Figure 2
(a) LenvatinibC21H19ClN4O4. (b) Cabozantinib C28H24FN3O5.
(a)(b)Theorem 2.
LetG1 and G2 be the graphs of lenvatinib and cabozantinib. Then, WPG1=48, and WPG2=62.Proof.
Since lenvatinib and cabozantinib are square-free graphs, we complete the proof by using Theorem1 and Table 1. Let Eij=uv∈EG1;du=i,dv=j such that EG1=∪1≤i<j≤3Eij:(6)WPG1=∑uv∈Eijdu−kuv−1dv−kuv−1−5NpG1−3NhG1,=∑uv∈Eij1≤i<j≤3kuv=0du−kuv−1dv−kuv−1+∑uv∈Eij1≤i<j≤3kuv=1du−kuv−1dv−kuv−1−33,=48.
In a similar way, we can complete the proof of cabozantinib.Table 1
Edge partition and number of hexagons.
GraphGEdge partition ofGiNh(1, 3)(2, 2)(2, 3)(3, 3)(1, 2)(3, 4)(2, 4)G1≅C21H19ClN4O4441951——3G2≅C28H24FN3O5372232224


### 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.Figure 3
(a) SilicateSL2; (b) silicate SL3.
(a)(b)Figure 4
(a) Dominating silicateDSL1; (b) dominating silicate DSL2.
(a)(b)Figure 5
(a) Regular triangulene silicateRTSL3; (b) regular triangulene silicate RTSL4.
(a)(b)The removal of silicon vertices (solid) from the silicate-related networks resulting with oxide type networks are shown in Figures6–8.Figure 6
(a) OxideOX2; (b) oxide OX3.
(a)(b)Figure 7
(a) Dominating oxideDOX1; (b) dominating oxide DOX2.
(a)(b)Figure 8
(a) Regular triangulene oxideRTOX3; (b) regular triangulene oxide RTOX4.
(a)(b)Recently,WP of oxide OXn and silicate SLn frameworks of dimension n have been computed in [12] using the third neighborhood of vertices as WpOXn=63n2−87n−3 and WpSLn=153n2−99n−3. Now, one can easily obtain these two results by putting k=1 and 2, respectively, in Corollary 1. We use DSLn, RTSLn, DOXn, and RTOXn 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.Table 2
Number of edges and Zagreb indices of silicate- and oxide-related structures.
GEGM1GM2GDSLn108n2−108n+361134n2−1242n+4322916n2−3456n+1242RTSLn [36]6n2+12n63n2+99n−18162n2+189n−72DOXn [31]54n2−54n+18432n2−480n+168864n2−1056n+384RTOXn [31]3n2+6n24n2+36n−848n2+48n−24We noted that the Zagreb indices of the dominating silicate network were computed with errors in [31] and can be readily corrected from Table 3.Table 3
Edge partition ofDSLn.
GEdge typeEidu,dv; uv∈EiEiDSLn1(3, 3)12n−62(3, 6)54n2−42n+123(6, 6)54n2−78n+30Theorem 3.
The Wiener polarity indices of silicate- and oxide-related structures are given by the following:(1)
WPDSLn=459n2−657n+249(2)
WPRTSLn=1/251n2+3n−36(3)
WPDOXn=189n2−267n+99(4)
WPRTOXn=1/221n2+3n−16Proof.
Since silicate- and oxide-related structures are pentagons-free chemical graphs, we haveNpG=0. Moreover, the end vertices of any edge in DSLn and RTSLn have exactly two common neighbors, whereas DOXn and RTOXn have one common neighbor. We can easily see that the number of hexagons in DSLn and DOXn are equal and the same truth for RTSLn and RTOXn structures i.e., NhDSLn=NhDOXn=9n2−15n+7 and NhRTSLn=NhRTOXn=nn−1/2. Based on the above values and by Table 2, we compute the Wiener polarity indices as follows:(7)WPDSLn=M2DSLn−3M1DSLn+9EDSLn−3NhDSLn,=2916n2−3456n+1242−31134n2−1242n+432+9108n2−108n+36−39n2−15n+7,=459n2−657n+249.WPRTSLn=M2RTSLn−3M1RTSLn+9ERTSLn−3NhRTSLn,=162n2+189n−72−363n2+99n−18+96n2+12n−39n2−15n+7,=1251n2+3n−36.WPDOXn=M2DOXn−2M1DOXn+4EDOXn−3NhDOXn,=864n2−1056n+384−3432n2−480n+168+954n2−54n+18−3nn−12,=189n2−267n+99.WPRTOXn=M2RTOXn−2M1RTOXn+4RTOXn−3NhRTOXn,=48n2+48n−24−324n2+36n−8+93n2+6n−3nn−12,=1221n2+3n−16.We now give the brief definitions of Sierpiński and its gasket graphs and followed by computations ofWP. The 1-dimensional Sierpiński graph S1 is a complete graph on three vertices, and an n-dimensional Sierpiński graph Sn is constructed by connecting three copies of Sn−1 with three bringing edges, as shown in Figure 9.The Sierpiński gasketSTn is obtained from Sn by contracting all its bridging edges (Figure 10). The first and second Zagreb indices of Sn and STn were computed in [37], and a,b-Zagreb index was dealt in [38]. The number of vertices and edges of Sn are 3n and 3n+1−3/2, respectively. Similarly, STn comprises of 3n+3/2 vertices and 3n edges.Figure 9
(a) Sierpiński graphS2; (b) Sierpiński graph S3.
(a)(b)Figure 10
(a) Sierpiński gasketST2; (b) Sierpiński gasket ST3.
(a)(b)Theorem 4.
Forn≥3, the Wiener polarity indices of Sierpiński graph and the Sierpiński gasket are given by the following:(1)
WPSn=8×3n−1−12(2)
WPSTn=26×3n−2−24Proof.
We complete the proof by using Theorem1 and Table 4. Now, we derive the first formula, and the second one can be obtained similarly. Let Eij=uv∈ESn;du=i,dv=j:(8)WPSn=∑uv∈ESndu−kuv−1dv−kuv−1−5NpSn−3NhSn,=∑uv∈E33kuv=0du−kuv−1dv−kuv−1+∑uv∈E33kuv=1du−kuv−1dv−kuv−1+∑uv∈E23kuv=1du−kuv−1dv−kuv−1−5NpSn−3NhSn,=43n−32+3n−6−33n−2,=8×3n−1−12.Table 4
The number of vertices, edges, pentagons, hexagons, and edge set partition ofSn and STn.
GNpGNhGEdge partitiondu,dv; uv∈EGkuvEijSn03n−2(3, 3)03n−3/213n−6(2, 3)16STn03n−3(4, 4)123n−1−323n−1(2, 4)16


### 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 hexafluorideSF6 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.Figure 11
Octahedral network. (a)OT1; (b) OT2.
(a)(b)Figure 12
Dominated octahedral network. (a)DOT1; (b) DOT2.
(a)(b)Figure 13
Rectangular octahedral network. (a)ROH3,9; (b) ROH4,8.
(a)(b)Theorem 5.
The Wiener polarity indices of octahedralOTn and dominating octahedral DOTn networks are given by the following:(1)
WPOTn=639n2−279n−3(2)
WPDOTn=1917n2−2475n+915Proof.
The Zagreb indices ofOTn have been obtained [40] as M1OTn=96n9n−1 and M2OTn=288n9n−2. Using Table 5, we can easily derive the Zagreb indices of DOTn as M1DOTn=2592n2−2784n+960 and M2DOTn=7776n2−8928n+3168. Moreover, the end vertices of any edge in OTn and DOTn have exactly two common neighbors, and we have(9)WPOTn=M2OTn−3M1OTn+9EOTn−3NhOTn,=288n9n−2−396n9n−1+972n2−33n2−3n+1,=639n2−279n−3.WPDOTn=M2DOTn−3M1DOTn+9EDOTn−3NhDOTn,=7776n2−8928n+3168−32592n2−2784n+960+9216n2−216n+72−39n2−15n+7,=1917n2−2475n+915.Table 5
Edge partition ofDOTn.
Gdu,dv; uv∈EGEi,jNhGDOTn(4, 4)54n2−30n+69n2−15n+7(8, 8)54n2−78n+30(4, 8)108n2−108n+36Theorem 6.
The Zagreb and Wiener polarity indices of rectangular octahedralROTm,n networks are given by the following:(1)
M1ROTm,n=144mn−32m−16n+16,:meven andnodd,144mn−32m−16n,:otherwise.(2)
M2ROTm,n=432mn−192m−96n+96,:meven andnodd,432mn−192m−96n+32,:otherwise.(3)
WPROTm,n=1/2213mn−186m−93n+90,:meven andnodd,1/2213mn−186m−93n+58,:otherwise.Proof.
ForROTm,n, m even and n odd, we have EROTm,n=E4,4∪E4,8∪E8,8 where E4,4=3mn+4m+2n−2, E4,8=6mn, and E8,8=3mn−4m−2n+2. In this case, the number of hexagon in ROTm,n is 1/2mn−2m−n+3, and the end vertices of any edge has exactly two common neighbors:(10)M1ROTm,n=∑uv∈E4,4du+dv+∑uv∈E4,8du+dv+∑uv∈E8,8du+dv,=83mn+4m+2n−2+126mn+163mn−4m−2n+2,=144mn−32m−16n+16.M2ROTm,n=∑uv∈E4,4du×dv+∑uv∈E4,8du×dv+∑uv∈E8,8du×dv,=163mn+4m+2n−2+326mn+643mn−4m−2n+2,=432mn−192m−96n+96.WPROTm,n=M2ROTm,n−k+1M1ROTm,n+k+12EROTm,n−3NhROTm,n,=432mn−192m−96n+96−3144mn−32m−16n+16+912mn−32mn−2m−n+3,=12213mn−186m−93n+90.
For all other values ofm and n in ROTm,n, we have E4,4=3mn+4m+2n+2, E4,8=6mn−4, E8,8=3mn−4m−2n+2, and NhROTm,n=1/2mn−2m−n+2. Then,(11)M1ROTm,n=∑uv∈E4,4du+dv+∑uv∈E4,8du+dv+∑uv∈E8,8du+dv,=83mn+4m+2n+2+126mn−4+163mn−4m−2n+2,=144mn−32m−16n.M2ROTm,n=∑uv∈E4,4du×dv+∑uv∈E4,8du×dv+∑uv∈E8,8du×dv,=163mn+4m+2n+2+326mn−4+643mn−4m−2n+2,=432mn−192m−96n+32.WPROTm,n=M2ROTm,n−k+1M1ROTm,n+k+12EROTm,n−3NhROTm,n,=432mn−192m−96n+32−3144mn−32m−16n+912mn−32mn−2m−n+2,=12213mn−186m−93n+58.We have shown the graphical plots of our computed results in Figures14 and 15.Figure 14
A graphical comparison of Wiener polarity indices of silicate, oxide, Sierpiński, and octahedral-derived structures.Figure 15
A graphical representation of the Wiener polarity index of rectangular octahedral networks.


## 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.Figure 2
(a) LenvatinibC21H19ClN4O4. (b) Cabozantinib C28H24FN3O5.
(a)(b)Theorem 2.
LetG1 and G2 be the graphs of lenvatinib and cabozantinib. Then, WPG1=48, and WPG2=62.Proof.
Since lenvatinib and cabozantinib are square-free graphs, we complete the proof by using Theorem1 and Table 1. Let Eij=uv∈EG1;du=i,dv=j such that EG1=∪1≤i<j≤3Eij:(6)WPG1=∑uv∈Eijdu−kuv−1dv−kuv−1−5NpG1−3NhG1,=∑uv∈Eij1≤i<j≤3kuv=0du−kuv−1dv−kuv−1+∑uv∈Eij1≤i<j≤3kuv=1du−kuv−1dv−kuv−1−33,=48.
In a similar way, we can complete the proof of cabozantinib.Table 1
Edge partition and number of hexagons.
GraphGEdge partition ofGiNh(1, 3)(2, 2)(2, 3)(3, 3)(1, 2)(3, 4)(2, 4)G1≅C21H19ClN4O4441951——3G2≅C28H24FN3O5372232224


## 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.Figure 3
(a) SilicateSL2; (b) silicate SL3.
(a)(b)Figure 4
(a) Dominating silicateDSL1; (b) dominating silicate DSL2.
(a)(b)Figure 5
(a) Regular triangulene silicateRTSL3; (b) regular triangulene silicate RTSL4.
(a)(b)The removal of silicon vertices (solid) from the silicate-related networks resulting with oxide type networks are shown in Figures6–8.Figure 6
(a) OxideOX2; (b) oxide OX3.
(a)(b)Figure 7
(a) Dominating oxideDOX1; (b) dominating oxide DOX2.
(a)(b)Figure 8
(a) Regular triangulene oxideRTOX3; (b) regular triangulene oxide RTOX4.
(a)(b)Recently,WP of oxide OXn and silicate SLn frameworks of dimension n have been computed in [12] using the third neighborhood of vertices as WpOXn=63n2−87n−3 and WpSLn=153n2−99n−3. Now, one can easily obtain these two results by putting k=1 and 2, respectively, in Corollary 1. We use DSLn, RTSLn, DOXn, and RTOXn 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.Table 2
Number of edges and Zagreb indices of silicate- and oxide-related structures.
GEGM1GM2GDSLn108n2−108n+361134n2−1242n+4322916n2−3456n+1242RTSLn [36]6n2+12n63n2+99n−18162n2+189n−72DOXn [31]54n2−54n+18432n2−480n+168864n2−1056n+384RTOXn [31]3n2+6n24n2+36n−848n2+48n−24We noted that the Zagreb indices of the dominating silicate network were computed with errors in [31] and can be readily corrected from Table 3.Table 3
Edge partition ofDSLn.
GEdge typeEidu,dv; uv∈EiEiDSLn1(3, 3)12n−62(3, 6)54n2−42n+123(6, 6)54n2−78n+30Theorem 3.
The Wiener polarity indices of silicate- and oxide-related structures are given by the following:(1)
WPDSLn=459n2−657n+249(2)
WPRTSLn=1/251n2+3n−36(3)
WPDOXn=189n2−267n+99(4)
WPRTOXn=1/221n2+3n−16Proof.
Since silicate- and oxide-related structures are pentagons-free chemical graphs, we haveNpG=0. Moreover, the end vertices of any edge in DSLn and RTSLn have exactly two common neighbors, whereas DOXn and RTOXn have one common neighbor. We can easily see that the number of hexagons in DSLn and DOXn are equal and the same truth for RTSLn and RTOXn structures i.e., NhDSLn=NhDOXn=9n2−15n+7 and NhRTSLn=NhRTOXn=nn−1/2. Based on the above values and by Table 2, we compute the Wiener polarity indices as follows:(7)WPDSLn=M2DSLn−3M1DSLn+9EDSLn−3NhDSLn,=2916n2−3456n+1242−31134n2−1242n+432+9108n2−108n+36−39n2−15n+7,=459n2−657n+249.WPRTSLn=M2RTSLn−3M1RTSLn+9ERTSLn−3NhRTSLn,=162n2+189n−72−363n2+99n−18+96n2+12n−39n2−15n+7,=1251n2+3n−36.WPDOXn=M2DOXn−2M1DOXn+4EDOXn−3NhDOXn,=864n2−1056n+384−3432n2−480n+168+954n2−54n+18−3nn−12,=189n2−267n+99.WPRTOXn=M2RTOXn−2M1RTOXn+4RTOXn−3NhRTOXn,=48n2+48n−24−324n2+36n−8+93n2+6n−3nn−12,=1221n2+3n−16.We now give the brief definitions of Sierpiński and its gasket graphs and followed by computations ofWP. The 1-dimensional Sierpiński graph S1 is a complete graph on three vertices, and an n-dimensional Sierpiński graph Sn is constructed by connecting three copies of Sn−1 with three bringing edges, as shown in Figure 9.The Sierpiński gasketSTn is obtained from Sn by contracting all its bridging edges (Figure 10). The first and second Zagreb indices of Sn and STn were computed in [37], and a,b-Zagreb index was dealt in [38]. The number of vertices and edges of Sn are 3n and 3n+1−3/2, respectively. Similarly, STn comprises of 3n+3/2 vertices and 3n edges.Figure 9
(a) Sierpiński graphS2; (b) Sierpiński graph S3.
(a)(b)Figure 10
(a) Sierpiński gasketST2; (b) Sierpiński gasket ST3.
(a)(b)Theorem 4.
Forn≥3, the Wiener polarity indices of Sierpiński graph and the Sierpiński gasket are given by the following:(1)
WPSn=8×3n−1−12(2)
WPSTn=26×3n−2−24Proof.
We complete the proof by using Theorem1 and Table 4. Now, we derive the first formula, and the second one can be obtained similarly. Let Eij=uv∈ESn;du=i,dv=j:(8)WPSn=∑uv∈ESndu−kuv−1dv−kuv−1−5NpSn−3NhSn,=∑uv∈E33kuv=0du−kuv−1dv−kuv−1+∑uv∈E33kuv=1du−kuv−1dv−kuv−1+∑uv∈E23kuv=1du−kuv−1dv−kuv−1−5NpSn−3NhSn,=43n−32+3n−6−33n−2,=8×3n−1−12.Table 4
The number of vertices, edges, pentagons, hexagons, and edge set partition ofSn and STn.
GNpGNhGEdge partitiondu,dv; uv∈EGkuvEijSn03n−2(3, 3)03n−3/213n−6(2, 3)16STn03n−3(4, 4)123n−1−323n−1(2, 4)16


## 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 hexafluorideSF6 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.Figure 11
Octahedral network. (a)OT1; (b) OT2.
(a)(b)Figure 12
Dominated octahedral network. (a)DOT1; (b) DOT2.
(a)(b)Figure 13
Rectangular octahedral network. (a)ROH3,9; (b) ROH4,8.
(a)(b)Theorem 5.
The Wiener polarity indices of octahedralOTn and dominating octahedral DOTn networks are given by the following:(1)
WPOTn=639n2−279n−3(2)
WPDOTn=1917n2−2475n+915Proof.
The Zagreb indices ofOTn have been obtained [40] as M1OTn=96n9n−1 and M2OTn=288n9n−2. Using Table 5, we can easily derive the Zagreb indices of DOTn as M1DOTn=2592n2−2784n+960 and M2DOTn=7776n2−8928n+3168. Moreover, the end vertices of any edge in OTn and DOTn have exactly two common neighbors, and we have(9)WPOTn=M2OTn−3M1OTn+9EOTn−3NhOTn,=288n9n−2−396n9n−1+972n2−33n2−3n+1,=639n2−279n−3.WPDOTn=M2DOTn−3M1DOTn+9EDOTn−3NhDOTn,=7776n2−8928n+3168−32592n2−2784n+960+9216n2−216n+72−39n2−15n+7,=1917n2−2475n+915.Table 5
Edge partition ofDOTn.
Gdu,dv; uv∈EGEi,jNhGDOTn(4, 4)54n2−30n+69n2−15n+7(8, 8)54n2−78n+30(4, 8)108n2−108n+36Theorem 6.
The Zagreb and Wiener polarity indices of rectangular octahedralROTm,n networks are given by the following:(1)
M1ROTm,n=144mn−32m−16n+16,:meven andnodd,144mn−32m−16n,:otherwise.(2)
M2ROTm,n=432mn−192m−96n+96,:meven andnodd,432mn−192m−96n+32,:otherwise.(3)
WPROTm,n=1/2213mn−186m−93n+90,:meven andnodd,1/2213mn−186m−93n+58,:otherwise.Proof.
ForROTm,n, m even and n odd, we have EROTm,n=E4,4∪E4,8∪E8,8 where E4,4=3mn+4m+2n−2, E4,8=6mn, and E8,8=3mn−4m−2n+2. In this case, the number of hexagon in ROTm,n is 1/2mn−2m−n+3, and the end vertices of any edge has exactly two common neighbors:(10)M1ROTm,n=∑uv∈E4,4du+dv+∑uv∈E4,8du+dv+∑uv∈E8,8du+dv,=83mn+4m+2n−2+126mn+163mn−4m−2n+2,=144mn−32m−16n+16.M2ROTm,n=∑uv∈E4,4du×dv+∑uv∈E4,8du×dv+∑uv∈E8,8du×dv,=163mn+4m+2n−2+326mn+643mn−4m−2n+2,=432mn−192m−96n+96.WPROTm,n=M2ROTm,n−k+1M1ROTm,n+k+12EROTm,n−3NhROTm,n,=432mn−192m−96n+96−3144mn−32m−16n+16+912mn−32mn−2m−n+3,=12213mn−186m−93n+90.
For all other values ofm and n in ROTm,n, we have E4,4=3mn+4m+2n+2, E4,8=6mn−4, E8,8=3mn−4m−2n+2, and NhROTm,n=1/2mn−2m−n+2. Then,(11)M1ROTm,n=∑uv∈E4,4du+dv+∑uv∈E4,8du+dv+∑uv∈E8,8du+dv,=83mn+4m+2n+2+126mn−4+163mn−4m−2n+2,=144mn−32m−16n.M2ROTm,n=∑uv∈E4,4du×dv+∑uv∈E4,8du×dv+∑uv∈E8,8du×dv,=163mn+4m+2n+2+326mn−4+643mn−4m−2n+2,=432mn−192m−96n+32.WPROTm,n=M2ROTm,n−k+1M1ROTm,n+k+12EROTm,n−3NhROTm,n,=432mn−192m−96n+32−3144mn−32m−16n+912mn−32mn−2m−n+2,=12213mn−186m−93n+58.We have shown the graphical plots of our computed results in Figures14 and 15.Figure 14
A graphical comparison of Wiener polarity indices of silicate, oxide, Sierpiński, and octahedral-derived structures.Figure 15
A graphical representation of the Wiener polarity index of rectangular octahedral networks.


## 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.



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