Wiener Polarity Index Calculation of Square-Free Graphs ...

Research ArticleWiener Polarity Index Calculation of Square-Free Graphs and ItsImplementation to Certain Complex Materials

Lin Zhang1 Tian-Le Sun 2 Micheal Arockiaraj 3 M Arulperumjothi 4

and S Prabhu 5

1School of Library Anhui Jianzhu University Hefei 230601 China2College of Economics Sichuan Agricultural University Chengdu 610000 China3Department of Mathematics Loyola College Chennai 600034 India4Department of Mathematics Loyola College University of Madras Chennai 600034 India5Department of Mathematics Sri Venkateswara College of Engineering Sriperumbudur Kanchipuram 602117 India

Correspondence should be addressed to S Prabhu drsavariprabhugmailcom

Received 23 October 2020 Revised 10 November 2020 Accepted 24 November 2020 Published 23 February 2021

Academic Editor Hijaz Ahmad

Copyright copy 2021 Lin Zhang et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Molecular topology is a portion ofmathematical chemistrymanaging the logarithmic portrayal of chemical materials permitting atremendous yet straightforward characterization of the compounds Concerning the traditional physical-chemical descriptors itis conceivable to set up direct quantitative structure-activity relationship methods to associate with such descriptors termedtopological indices In this study we have developed the mathematical technique to study the Wiener polarity index of chemicalmaterials without squares We have taken the cancer treatment drugs such as lenvatinib and cabozantinib to illustrate ourapproach In addition we explored the inherent property of silicate Sierpinski and octahedral-related complex materials that theedge set can be decomposed in such a way that any edge in the same part of the decomposition has an equal number ofneighboring vertices and applied the technique to derive the formulae for these materials

1 Introduction

Drug discovery is the procedure through which potentialnew helpful substances are established by utilizing a com-bination of computational experimental translational andclinical models Regardless of advances in biotechnology andcomprehension of organic frameworks drug discovery is asyet an extensive costly complicated and inefficient processwith a high attrition rate of new therapeutic discovery Drugdesign is the inventive process of finding new drugsdepending on the information of a biological target Cur-rently there are several incredible methodologies for drugdesign and drug database screening [1ndash3] +e quantitativestructure activity and property relationships (QSARQSPR)are mathematical models that endeavor to transmit thestructure-derived features of a compound to its biological orphysico-chemical movement +e relevance of QSARQSPRadvance in scientific research starts with the idea of the

assortment of the input data from databases then the ex-planation of molecular descriptors which describes impor-tant information of the composed molecules and throughan explicit method to decrease the number of descriptors tothe most instructive descriptors and the last part is thechemoinformatic tools which are used to assemble modelsthat describe the pragmatic relationship between thestructure and property or activity

A significant development in the automated computertreatment of chemical materials and QSAR has been theapplication of a numerical procedure namely graph theoryto chemistry In chemical graph theory molecular structuresare represented as hydrogen-suppressed graphs regularlyknown as molecular graphs in which the atoms are taken byvertices and the bonds by edges +e associations betweenmolecules can be portrayed by different kinds of topologicalmatrices for example distance or adjacency matrices whichcan be scientifically converted into real numbers and called

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 6686352 13 pageshttpsdoiorg10115520216686352

topological indices Topological indices (TIs) are usuallyconsidered the atomic arrangement of compounds such asatomic size shape branching nearness of heteroatoms andvarious bonds [6ndash10] +e handiness of TIs in QSPR andQSAR contemplates has been broadly illustrated [4] andthey likewise have been utilized as a proportion of auxiliarystructural similarity or diversity by their application todatabases created by PC +e idea of topological indicesoriginated from the work of Wiener while he was func-tioning on the paraffin boiling points using Wiener andWiener polarity indices [5]

All the graphs argued in this paper are simple and finiteLet G (V(G) E(G)) be a connected graph +e cardinalityof the vertex set and the edge set of G is represented by|V(G)| and |E(G)| respectively +e number of pentagonsand hexagons of G is denoted by Np(G) and Nh(G) re-spectively For any positive integer i we represent Ni

G(v)

u isin V(G) dG(u v) i1113864 1113865 as the ith neighborhood of v andthus clearly the open neighborhood of v (denoted by NG(v))is N1

G(v) and the degree of v (denoted by dv) is |N1G(v)| +e

first and second Zagreb indices of a graph G are definedrespectively as follows

M1(G) 1113944uvisinE(G)

du + dv1113858 1113859 1113944visinV(G)

d2v


du times dv1113858 1113859(1)

+e Wiener polarity index of a graph G is defined as

WP(G) u v dG(u v) 3 u v isin V(G)1113864 11138651113868111386811138681113868

1113868111386811138681113868 (2)

It was observed thatWP(G) (12)1113936visinV(G)|N3G(v)| [11]

and the importance of WP has been demonstrated in variouspapers [9 10 12ndash28] In this paper we study the Wienerpolarity index of C4-free graphs and implement our tech-nique to cancer treatment drugs silicate Sierpinski andoctahedral-related networks

2 Derivation of the Key Result

For a tree T it was realized [18] thatWP(T) 1113936uvisinE(T)(du minus 1) times (dv minus 1) and interms ofZagreb indices can be rewritten asWP(T) M2(T) minus M1(T) + |E(T)| For a graph G where G

is C3-free and C4-free such that its different cycles have atmost one common edge [14] WP(G) M2(G)minus

M1(G) + |E(G)| minus 5NP(G) minus 3Nh(G) Suppose G is theC3-free graph such that its different cycles have at most onecommon edge [29] WP(G) M2(G)minus

M1(G) + |E(G)| minus 5Np(G) minus 3Nh(G) minus 1113936i(Ki minus 4)Nqi(G)

where Nqi(G) is the number of quadrangles of type i such

that the sum of degrees on the vertices of that type quad-rangle is Ki In this section we fill a gap by computingWiener polarity indices of C4-free graphs

Theorem 1 Let G be a C4-free graph such that its differentcycles have at most one common edge For uv isin E(G) letkuv |NG(u)capNG(v)| 3en

WP(G) 1113944uvisinE(G)

du minus kuv minus 1( 1113857 times dv minus kuv minus 1( 1113857

minus 5Np(G) minus 3Nh(G)

(3)

Proof Suppose that for any edge uv of G that kuv 0 it iseasy to see that [14 29] WP(G) 1113936uvisinE(G)(du minus 1) times

(dv minus 1) minus 5NP(G) minus 3Nh(G) Assume that there exists anedge uv isin E(G) such that kuv ne 0 as shown in Figure 1+enthe number of pairs of vertices x y1113864 1113865 of G such thatdG(x y) 3 and containing the edge uv as the internal edgeis | (xi yj) 1le ilep 1le jle q1113966 1113967| Since du p + kuv + 1 anddv q + kuv + 1 we have pq (du minus kuv minus 1)(dv minus kuv minus 1)Hence


du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857 minus 5Np(G) minus 3Nh(G)

(4)

Corollary 1 Let G be a C4-free graph such that its differentcycles have at most one common edge If every edge uv isin E(G)

such that kuv kge 0 then

WP(G) M2(G) minus (k + 1)M1(G) +(k + 1)2|E(G)|

minus 5Np(G) minus 3Nh(G)(5)

3 Numerical Computation

In this section we compute the numerical quantities forWiener polarity indices of lenvatinib and cabozantinibcancer drugs and chemical materials based on the silicateSierpinski and octahedral frameworks

31 Cancer Treatment Drugs Lenvatinib and cabozantinib[30] are orally accessible multikinase inhibitor and anti-neoplastic specialist that are utilized in the treatment ofadvanced metastatic medullary thyroid cancer (MTC) andhepatocellular carcinoma (HCC) Lenvatinib is a member ofthe class of quinolines that is the carboxamide of 4-3-chloro-4-[(cyclopropylcarbamoyl)amino]phenoxy-7-methoxyquinoline-6-carboxylic acid Cabozantinib is a di-carboxylic acid diamide that is N-phenyl-Nrsquo-(4-fluo-rophenyl)cyclopropane-11-dicarboxamide in which thehydrogen at position 4 on the phenyl ring is substituted byan (67-dimethoxyquinolin-4-yl)oxy group +e moleculargraph structures of lenvatinib and cabozantinib are shown inFigure 2

Theorem 2 Let G1 and G2 be the graphs of lenvatinib andcabozantinib 3en WP(G1) 48 and WP(G2) 62

Proof Since lenvatinib and cabozantinib are square-freegraphs we complete the proof by using +eorem 1 andTable 1 Let Eij uv isin E(G1) du i dv j1113864 1113865 such thatE(G1) cup 1leiltjle3Eij

2 Mathematical Problems in Engineering

WP G1( 1113857 1113944uvisinEij

du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857 minus 5Np G1( 1113857 minus 3Nh G1( 1113857

1113944uvisinEij

1leiltjle3kuv0

du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857 + 1113944uvisinEij

1leiltjle3kuv1

du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857 minus 3(3)

48

(6)

H2N

O

N

O

ONH

HN

O

CH3

Cl

(a)

CH3

O

O

CH3

N

O

NH

OO

NH F

(b)

Figure 2 (a) Lenvatinib C21H19ClN4O4 (b) Cabozantinib C28H24FN3O5

Table 1 Edge partition and number of hexagons

Graph GEdge partition of Gi

Nh(1 3) (2 2) (2 3) (3 3) (1 2) (3 4) (2 4)

G1 C21H19ClN4O4 4 4 19 5 1 mdash mdash 3G2 C28H24FN3O5 3 7 22 3 2 2 2 4

w1

w2

x1 y1

y2

y3

x2

x3

u v

wkuv

xpndash1yqndash1

xp yq

Figure 1 Edge uv of G such that kuv ne 0

Mathematical Problems in Engineering 3

In a similar way we can complete the proof ofcabozantinib

32 Silicate andSierpinskiGraphs Inorganic networks basedon silicates and fractal types three-dimensional metal-cat-echolate frameworks metal-organic frameworks and re-ticular chemistry as a whole are emerging as cutting-edgefields of research in catalysis and ultrahigh proton con-ductivity +e silicate-related [12 31 32] and Imran Sabeel-E-Hafi [33ndash35] networks are depicted in Figures 3ndash5

+e removal of silicon vertices (solid) from the silicate-related networks resulting with oxide type networks areshown in Figures 6ndash8

Recently WP of oxide OX(n) and silicate SL(n) frame-works of dimension n have been computed in [12] using thethird neighborhood of vertices as Wp(OX(n)) 63n2 minus 87n minus

3 and Wp(SL(n)) 153n2 minus 99n minus 3 Now one can easilyobtain these two results by putting k 1 and 2 respectively inCorollary 1 We use DSL(n) RTSL(n) DOX(n) andRTOX(n) to represent the n-dimensional dominating silicateregular triangulene silicate dominating oxide and regulartriangulene oxide networks respectively We now recall theZagreb indices of silicate- and oxide-related structures inTable 2

We noted that the Zagreb indices of the dominatingsilicate network were computed with errors in [31] and canbe readily corrected from Table 3

Theorem 3 3e Wiener polarity indices of silicate- andoxide-related structures are given by the following

(1) WP(DSL(n)) 459n2 minus 657n + 249(2) WP(RTSL(n)) (12)(51n2 + 3n minus 36)

(3) WP(DOX(n)) 189n2 minus 267n + 99(4) WP(RTOX(n)) (12)(21n2 + 3n minus 16)

Proof Since silicate- and oxide-related structures are pen-tagons-free chemical graphs we have Np(G) 0 Moreoverthe end vertices of any edge in DSL(n) and RTSL(n) haveexactly two common neighbors whereas DOX(n) andRTOX(n) have one common neighbor We can easily seethat the number of hexagons in DSL(n) and DOX(n) areequal and the same truth for RTSL(n) and RTOX(n)

structures ie Nh(DSL(n)) Nh(DOX(n)) 9n2 minus 15n + 7and Nh(RTSL(n)) Nh(RTOX(n)) n(n minus 1)2 Based onthe above values and by Table 2 we compute the Wienerpolarity indices as follows

WP(DSL(n)) M2(DSL(n)) minus 3M1(DSL(n)) + 9|E(DSL(n))| minus 3Nh(DSL(n))

2916n2

minus 3456n + 1242 minus 3 1134n2

minus 1242n + 4321113872 1113873 + 9 108n2

minus 108n + 361113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

459n2

minus 657n + 249

WP(RTSL(n)) M2(RTSL(n)) minus 3M1(RTSL(n)) + 9|E(RTSL(n))| minus 3Nh(RTSL(n))

162n2

+ 189n minus 72 minus 3 63n2

+ 99n minus 181113872 1113873 + 9 6n2

+ 12n1113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

12

51n2

+ 3n minus 361113872 1113873

WP(DOX(n)) M2(DOX(n)) minus 2M1(DOX(n)) + 4|E(DOX(n))| minus 3Nh(DOX(n))

864n2

minus 1056n + 384 minus 3 432n2

minus 480n + 1681113872 1113873 + 9 54n2

minus 54n + 181113872 1113873 minus3n(n minus 1)

2

189n2

minus 267n + 99

WP(RTOX(n)) M2(RTOX(n)) minus 2M1(RTOX(n)) + 4|(RTOX(n))| minus 3Nh(RTOX(n))

48n2


+ 36n minus 81113872 1113873 + 9 3n2

+ 6n1113872 1113873 minus3n(n minus 1)

2

12

21n2

+ 3n minus 161113872 1113873

(7)

We now give the brief definitions of Sierpinski and itsgasket graphs and followed by computations of WP +e1-dimensional Sierpinski graph S1 is a complete graph on

three vertices and an n-dimensional Sierpinski graph Sn isconstructed by connecting three copies of Snminus 1 with threebringing edges as shown in Figure 9


+e Sierpinski gasket STn is obtained from Sn by con-tracting all its bridging edges (Figure 10)+e first and secondZagreb indices of Sn and STn were computed in [37] and(a b)-Zagreb index was dealt in [38] +e number of verticesand edges of Sn are 3n and (3n+1 minus 3)2 respectively SimilarlySTn comprises of (3n + 3)2 vertices and 3n edges

Theorem 4 For nge 3 the Wiener polarity indices of Sier-pinski graph and the Sierpinski gasket are given by thefollowing

(1) WP(Sn) 8 times 3nminus 1 minus 12(2) WP(STn) 26 times 3nminus 2 minus 24

(a) (b)

Figure 3 (a) Silicate SL(2) (b) silicate SL(3)

(a) (b)

Figure 4 (a) Dominating silicate DSL(1) (b) dominating silicate DSL(2)


Proof We complete the proof by using +eorem 1 andTable 4 Now we derive the first formula and the second one

can be obtained similarly Let Eij uv isin E(Sn) du 1113864

i dv j

WP Sn( 1113857 1113944

uvisinE Sn( )

du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857 minus 5Np Sn( 1113857 minus 3Nh Sn( 1113857

1113944uvisinE33kuv0

du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857 + 1113944uvisinE33kuv1

du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857

middot + 1113944uvisinE23kuv1


43n

minus 32

1113888 1113889 + 3nminus 6 minus 3 3nminus 2

1113872 1113873

8 times 3nminus 1minus 12

(8)

(a) (b)

Figure 5 (a) Regular triangulene silicate RTSL(3) (b) regular triangulene silicate RTSL(4)

(a) (b)

Figure 6 (a) Oxide OX(2) (b) oxide OX(3)


Table 2 Number of edges and Zagreb indices of silicate- and oxide-related structures

G E(G) M1(G) M2(G)

DSL(n) 108n2 minus 108n + 36 1134n2 minus 1242n + 432 2916n2 minus 3456n + 1242RTSL(n) [36] 6n2 + 12n 63n2 + 99n minus 18 162n2 + 189n minus 72DOX(n) [31] 54n2 minus 54n + 18 432n2 minus 480n + 168 864n2 minus 1056n + 384RTOX(n) [31] 3n2 + 6n 24n2 + 36n minus 8 48n2 + 48n minus 24

Table 3 Edge partition of DSL(n)

G Edge type Ei (du dv) uv isin Ei Ei

DSL(n)

1 (3 3) 12n minus 62 (3 6) 54n2 minus 42n + 123 (6 6) 54n2 minus 78n + 30

(a) (b)

Figure 8 (a) Regular triangulene oxide RTOX(3) (b) regular triangulene oxide RTOX(4)

(a) (b)

Figure 7 (a) Dominating oxide DOX(1) (b) dominating oxide DOX(2)


33 Octahedral Structures +e idea of octahedral coordi-nation geometry was created by Alfred Werner in 1913 forwhich he was awarded the Nobel Prize in Chemistry [39] Heclarified the stoichiometries and isomerism in the coordi-nation mix using octahedral coordination geometry Hisunderstanding permitted scientists to legitimize the number

of isomers of coordination mixes Octahedral progress metalbuildings containing amines and basic anions are regularlyalluded to as Werner-type edifices

In chemistry octahedral molecular geometry portraysthe shape of compounds with six atoms or gatherings ofatoms or ligands symmetrically arranged around a focal

(a) (b)

Figure 10 (a) Sierpinski gasket ST2 (b) Sierpinski gasket ST3

(a) (b)

Figure 9 (a) Sierpinski graph S2 (b) Sierpinski graph S3

Table 4 +e number of vertices edges pentagons hexagons and edge set partition of Sn and STn

G Np(G) Nh(G) Edge partition (du dv) uv isin E(G) kuv |Eij|

Sn 0 3nminus 2 (3 3) 0 (3n minus 3)21 3n minus 6

(2 3) 1 6

STn 0 3nminus 3 (4 4) 1 2(3nminus 1 minus 3)

2 3nminus 1

(2 4) 1 6


atom characterizing the vertices of an octahedron +eoctahedron is one of the platonic solids even though oc-tahedral molecules commonly have an atom in their centreand no bonds between the ligand atoms A perfect octa-hedron fits the point group OH Illustrations of octahedralcompounds are sulfur hexafluoride SF6 and molybdenum

hexacarbonyl Mo(CO)6 +e term octahedral is usedsomewhat lightly by chemists concentrating on the ge-ometry of the bonds to the central atom and not consideringmodifications between the ligands themselves

+e octahedral network was very recently introduced in[40] Here we extend the network to its rectangular form

(a) (b)

Figure 12 Dominated octahedral network (a) DOT(1) (b) DOT(2)

(a) (b)

Figure 11 Octahedral network (a) OT(1) (b) OT(2)


0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n

DSL (n)RTSL (n)DOX (n)

RTOX (n)SnSTn

OT (n)DOT (n)

times104

Figure 14 A graphical comparison of Wiener polarity indices of silicate oxide Sierpinski and octahedral-derived structures

1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP

Figure 15 A graphical representation of the Wiener polarity index of rectangular octahedral networks

(a) (b)

Figure 13 Rectangular octahedral network (a) ROH(3 9) (b) ROH(4 8)

Table 5 Edge partition of DOT(n)

G (du dv) uv isin E(G) |Eij| Nh(G)

DOT(n)

(4 4) 54n2 minus 30n + 69n2 minus 15n + 7(8 8) 54n2 minus 78n + 30

(4 8) 108n2 minus 108n + 36


with the help of the idea adopted in [41] +e octahedralnetwork and its dominated version are depicted respec-tively in Figures 11 and 12 whereas the rectangular form isportrayed in Figure 13

Theorem 5 3e Wiener polarity indices of octahedralOT(n) and dominating octahedral DOT(n) networks aregiven by the following

(1) WP(OT(n)) 639n2 minus 279n minus 3

(2) WP(DOT(n)) 1917n2 minus 2475n + 915

Proof +e Zagreb indices of OT(n) have been obtained [40]as M1(OT(n)) 96n(9n minus 1) and M2(OT(n)) 288n

(9n minus 2) Using Table 5 we can easily derive the Zagrebindices of DOT(n) as M1(DOT(n)) 2592n2 minus 2784n +

960 and M2(DOT(n)) 7776n2 minus 8928n + 3168 Moreoverthe end vertices of any edge in OT(n) and DOT(n) haveexactly two common neighbors and we have

WP(OT(n)) M2(OT(n)) minus 3M1(OT(n)) + 9|E(OT(n))| minus 3Nh(OT(n))

288n(9n minus 2) minus 3(96n(9n minus 1)) + 9 72n2

1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3

WP(DOT(n)) M2(DOT(n)) minus 3M1(DOT(n)) + 9|E(DOT(n))| minus 3Nh(DOT(n))

7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)

Theorem 6 3e Zagreb and Wiener polarity indices ofrectangular octahedral ROT(m n) networks are given by thefollowing

(1) M1(ROT(m n)) 144mn minus 321113864 m minus 16n + 16

m even and n odd 144mn minus 32m minus 16n otherwise(2) M2(ROT(m n)) 432mn minus 192m minus 96n + 96 1113864

m even and n odd 432mn minus 192m minus 96n +

32 otherwise

(3) WP(ROT(m n)) (12)(213mn minus 1861113864 m minus 93n +

90) m even and n odd (12)(213mn minus 186m minus

93n + 58) otherwise

Proof For ROT(m n) m even and n odd we haveE(ROT(m n)) E44 cupE48 cupE88 where |E44| 3mn + 4m+

2n minus 2 |E48| 6mn and |E88| 3mn minus 4m minus 2n + 2 In thiscase the number of hexagon in ROT(m n) is(12)(mn minus 2m minus n + 3) and the end vertices of any edge hasexactly two common neighbors

M1(ROT(m n)) 1113944uvisinE44

du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857

8(3mn + 4m + 2n minus 2) + 12(6mn) + 16(3mn minus 4m minus 2n + 2)

144mn minus 32m minus 16n + 16


du times dv( 1113857 + 1113944uvisinE48


du times dv( 1113857



WP(ROT(m n)) M2(ROT(m n)) minus (k + 1)M1(ROT(m n)) +(k + 1)2|E(ROT(m n))| minus 3Nh(ROT(m n))

432mn minus 192m minus 96n + 96 minus 3(144mn minus 32m minus 16n + 16) + 9(12mn) minus32

(mn minus 2m minus n + 3)

12

(213mn minus 186m minus 93n + 90)

(10)


For all other values of m and n in ROT(m n) we have|E44| 3mn + 4m + 2n + 2 |E48| 6mn minus 4 |E88| 3mn minus

4m minus 2n + 2 and Nh(ROT(m n)) (12)(mn minus 2m minus n +

2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857

8(3mn + 4m + 2n + 2) + 12(6mn minus 4) + 16(3mn minus 4m minus 2n + 2)

144mn minus 32m minus 16n








432mn minus 192m minus 96n + 32 minus 3(144mn minus 32m minus 16n) + 9(12mn) minus32


12


(11)

We have shown the graphical plots of our computedresults in Figures 14 and 15

4 Conclusion

In this paper we have derived the technique to find theWiener polarity indices of graphs without squares andconsequently we have computed the Wiener polarityindices of chemical structures of lenvatinib and cabo-zantinib which are used in the treatment of thyroid cancerand HCC As measured topological indices are proficientat forecasting different properties and behaviors such asboiling point entropy enthalpy and critical pressure ourresults can be useful in designing new drugs and vaccinesfor cancer In addition to this we have computed theWiener polarity indices of some special classes of graphsnamely silicate Sierpinski and octahedral structures withthe help of our extended result

Data Availability

+e figures tables and other data used to support thefindings of this study are included within the article

Conflicts of Interest

+e authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S-F Zhou and W-Z Zhong ldquoDrug design and discoveryprinciples and applicationsrdquo Molecules vol 22 no 2 p 2792017

[2] K Balasubramanian ldquoMathematical and computationaltechniques for drug discovery promises and developmentsrdquoCurrent Topics in Medicinal Chemistry vol 18 no 32pp 2774ndash2799 2018

[3] W Gao Y Wang B Basavanagoud and M K JamilldquoCharacteristics studies of molecular structures in drugsrdquoSaudi Pharmaceutical Journal vol 25 no 4 pp 580ndash5862017

[4] J Devillers and A T Balaban Topological Indices and RelatedDescriptors in QSAR and QSPR Gordon amp Breach Amster-dam Netherlands 1999

[5] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947

[6] J B Liu J Zhao and J Min ldquoOn the Hosoya index of graphsformed by a fractal graphrdquo Fractals vol 27 no 8 pp 19ndash352019

[7] J-B Liu J Zhao H He and Z Shao ldquoValency-based to-pological descriptors and structural property of the gener-alized Sierpinski networksrdquo Journal of Statistical Physicsvol 177 no 6 pp 1131ndash1147 2019

[8] J B Liu J Zhao and Z Cai ldquoOn the generalized adjacencyLaplacian and signless Laplacian spectra of the weighted edgecorona networksrdquo Physica A vol 540 pp 12ndash30 2020

[9] W Fang M Ma F Chen and H Dong ldquo+ird smallestWiener polarity index of unicyclic graphsrdquo Frontiers inPhysics vol 8 2020


[10] M Alaeiyan F Afzal M R Farahani andM A Rostami ldquoAnexact formulas for the Wiener polarity index of nanostardendrimersrdquo Journal of Information and Optimization Sci-ences vol 41 no 4 p 933 2020

[11] L Chen T Li J Liu Y Shi and H Wang ldquoOn the Wienerpolarity index of lattice networksrdquo PLoS One vol 11 no 12Article ID e0167075 2016

[12] M Arockiaraj S R J Kavitha K Balasubramanian andI Gutman ldquoHyper-Wiener and Wiener polarity indices ofsilicate and oxide frameworksrdquo Journal of MathematicalChemistry vol 56 no 5 pp 1493ndash1510 2018

[13] A R Ashrafi and A Ghalavand ldquoOrdering chemical trees byWiener polarity indexrdquo Applied Mathematics and Compu-tation vol 313 pp 301ndash312 2017

[14] A Behmaram H Yousefi-Azari and A R Ashrafi ldquoWienerpolarity index of fullerenes and hexagonal systemsrdquo AppliedMathematics Letters vol 25 no 10 pp 1510ndash1513 2012

[15] H Deng H Xiao and F Tang ldquoOn the extremal Wienerpolarity index of trees with a given diameterrdquo MATCHCommunications in Mathematical and in Computer Chem-istry vol 63 no 1 pp 257ndash264 2010

[16] H Deng and H Xiao ldquo+e Wiener polarity index of mo-lecular graphs of alkanes with a given number of methylgroupsrdquo Journal of the Serbian Chemical Society vol 75no 10 pp 1405ndash1412 2010

[17] H Deng ldquoOn the extremal Wiener polarity index of chemicaltreesrdquo MATCH Communications in Mathematical and inComputer Chemistry vol 66 no 1 pp 305ndash314 2011

[18] W Du X Li and Y Shi ldquoAlgorithms and extremal problemon Wiener polarity indexrdquo MATCH Communications inMathematical and in Computer Chemistry vol 62 no 1pp 235ndash244 2009

[19] H Hosoya ldquoMathematical and chemical analysis of Wienerrsquospolarity numberrdquo in Topology in Chemistry Discrete Math-ematics of Molecules D H Rouvray and R B King EdsHorwood Chichester UK 2002

[20] H Hou B Liu and Y Huang ldquo+e maximum Wiener po-larity index of unicyclic graphsrdquo Applied Mathematics andComputation vol 218 no 20 pp 10149ndash10157 2012

[21] H Hua and K C Das ldquoOn the Wiener polarity index ofgraphsrdquo Applied Mathematics and Computation vol 280pp 162ndash167 2016

[22] A Ilic and M Ilic ldquoGeneralizations of Wiener polarity indexand terminal Wiener indexrdquo Graphs and Combinatoricsvol 29 no 5 pp 1403ndash1416 2013

[23] G Liu and G Liu ldquoWiener polarity index of dendrimersrdquoApplied Mathematics and Computation vol 322 pp 151ndash1532018

[24] B Liu H Hou and Y Huang ldquoOn the Wiener polarity indexof trees with maximum degree or given number of leavesrdquoComputers amp Mathematics with Applications vol 60 no 7pp 2053ndash2057 2010

[25] I Lukovits and W Linert ldquoPolarity-numbers of cycle-con-taining structuresrdquo Journal of Chemical Information andComputer Sciences vol 38 no 4 pp 715ndash719 1998

[26] J Ma Y Shi and J Yue ldquo+e Wiener polarity index of graphproductsrdquo Ars Combinatoria vol 116 pp 235ndash244 2014

[27] J Ma Y Shi Z Wang and J Yue ldquoOn Wiener polarity indexof bicyclic networksrdquo Scientific Reports vol 6 Article ID19066 2016

[28] Y Zhang and Y Hu ldquo+e Nordhaus-Gaddum-type inequalityfor the Wiener polarity indexrdquo Applied Mathematics andComputation vol 273 pp 880ndash884 2016

[29] M Arockiaraj J B Liu S Prabhu and M ArulperumjothildquoOn the zagreb and wiener polarity indices of C3-freechemical nanostructuresrdquo Utilitas Mathematica 2018

[30] D Li S Sedano R Allen J Gong M Cho and S SharmaldquoCurrent treatment landscape for advanced hepatocellularcarcinoma patient outcomes and the impact on quality ofliferdquo Cancers vol 11 no 6 p 841 2019

[31] A Q Baig M Imran and H Ali ldquoOn topological indices ofpoly oxide poly silicate DOX and DSL networksrdquo CanadianJournal of Chemistry vol 93 no 7 pp 730ndash739 2015

[32] M Cancan D Afzal S Hussain A Maqbool and F AfzalldquoSome new topological indices of silicate network viaM-polynomialrdquo Journal of Discrete Mathematical Sciencesand Cryptography vol 23 no 6 pp 1157ndash1171 2020

[33] M Imran Sabeel-E-Hafi W Gao and M Reza Farahani ldquoOntopological properties of sierpinski networksrdquo Chaos Solitonsamp Fractals vol 98 pp 199ndash204 2017

[34] J A Rodrıguez-Velazquez and J Tomas-Andreu ldquoOn theRandic index of polymer networks modelled by generalizedSierpinskirdquo MATCH Communications in Mathematical andin Computer Chemistry vol 74 no 1 pp 145ndash160 2015

[35] S Ediz M Alaeiyan M Alaeiyan M Farahani andM Cancan ldquoOn Van r and s topological properties of theSierpinski triangle networksrdquo Eurasian Chemical Commu-nications vol 2 no 7 p 819 2020

[36] M Imran A Q Baig H Ali and S U Rehman ldquoOn to-pological properties of poly honeycomb networksrdquo PeriodicaMathematica Hungarica vol 73 no 1 pp 100ndash119 2016

[37] H M A Siddiqui ldquoComputation of Zagreb indices andZagreb polynomials of Sierpinski graphsrdquo Hacettepe Journalof Mathematics and Statistics vol 49 no 2 pp 754ndash765 2020

[38] P Sarkar N De I N Cangul and A Pal ldquo+e (a b)-Zagrebindex of some derived networksrdquo Journal of Taibah Universityfor Science vol 13 no 1 pp 79ndash86 2020

[39] E C Constable and C E Housecroft ldquoCoordinationchemistry the scientific legacy of Alfred Wernerrdquo ChemicalSociety Reviews vol 42 no 4 pp 1429ndash1439 2013

[40] M Arockiaraj S R J Kavitha K Balasubramanian andJ B Liu ldquoOn certain topological indices of octahedral andicosahedral networksrdquo IET Control 3eory amp Applicationsvol 12 no 2 pp 215ndash220 2018

[41] J-B Liu M K Shafiq H Ali A Naseem N Maryam andS S Asghar ldquoTopological indices of mth chain silicategraphsrdquo Mathematics vol 7 no 1 p 42 2019


topological indices Topological indices (TIs) are usuallyconsidered the atomic arrangement of compounds such asatomic size shape branching nearness of heteroatoms andvarious bonds [6ndash10] +e handiness of TIs in QSPR andQSAR contemplates has been broadly illustrated [4] andthey likewise have been utilized as a proportion of auxiliarystructural similarity or diversity by their application todatabases created by PC +e idea of topological indicesoriginated from the work of Wiener while he was func-tioning on the paraffin boiling points using Wiener andWiener polarity indices [5]

All the graphs argued in this paper are simple and finiteLet G (V(G) E(G)) be a connected graph +e cardinalityof the vertex set and the edge set of G is represented by|V(G)| and |E(G)| respectively +e number of pentagonsand hexagons of G is denoted by Np(G) and Nh(G) re-spectively For any positive integer i we represent Ni

G(v)

u isin V(G) dG(u v) i1113864 1113865 as the ith neighborhood of v andthus clearly the open neighborhood of v (denoted by NG(v))is N1

G(v) and the degree of v (denoted by dv) is |N1G(v)| +e

first and second Zagreb indices of a graph G are definedrespectively as follows


du + dv1113858 1113859 1113944visinV(G)

d2v


du times dv1113858 1113859(1)

+e Wiener polarity index of a graph G is defined as

WP(G) u v dG(u v) 3 u v isin V(G)1113864 11138651113868111386811138681113868

1113868111386811138681113868 (2)

It was observed thatWP(G) (12)1113936visinV(G)|N3G(v)| [11]

and the importance of WP has been demonstrated in variouspapers [9 10 12ndash28] In this paper we study the Wienerpolarity index of C4-free graphs and implement our tech-nique to cancer treatment drugs silicate Sierpinski andoctahedral-related networks

2 Derivation of the Key Result

For a tree T it was realized [18] thatWP(T) 1113936uvisinE(T)(du minus 1) times (dv minus 1) and interms ofZagreb indices can be rewritten asWP(T) M2(T) minus M1(T) + |E(T)| For a graph G where G

is C3-free and C4-free such that its different cycles have atmost one common edge [14] WP(G) M2(G)minus

M1(G) + |E(G)| minus 5NP(G) minus 3Nh(G) Suppose G is theC3-free graph such that its different cycles have at most onecommon edge [29] WP(G) M2(G)minus

M1(G) + |E(G)| minus 5Np(G) minus 3Nh(G) minus 1113936i(Ki minus 4)Nqi(G)

where Nqi(G) is the number of quadrangles of type i such

that the sum of degrees on the vertices of that type quad-rangle is Ki In this section we fill a gap by computingWiener polarity indices of C4-free graphs

Theorem 1 Let G be a C4-free graph such that its differentcycles have at most one common edge For uv isin E(G) letkuv |NG(u)capNG(v)| 3en


du minus kuv minus 1( 1113857 times dv minus kuv minus 1( 1113857

minus 5Np(G) minus 3Nh(G)

(3)

Proof Suppose that for any edge uv of G that kuv 0 it iseasy to see that [14 29] WP(G) 1113936uvisinE(G)(du minus 1) times

(dv minus 1) minus 5NP(G) minus 3Nh(G) Assume that there exists anedge uv isin E(G) such that kuv ne 0 as shown in Figure 1+enthe number of pairs of vertices x y1113864 1113865 of G such thatdG(x y) 3 and containing the edge uv as the internal edgeis | (xi yj) 1le ilep 1le jle q1113966 1113967| Since du p + kuv + 1 anddv q + kuv + 1 we have pq (du minus kuv minus 1)(dv minus kuv minus 1)Hence


du minus kuv minus 1( 1113857 dv minus kuv minus 1( 1113857 minus 5Np(G) minus 3Nh(G)

(4)

Corollary 1 Let G be a C4-free graph such that its differentcycles have at most one common edge If every edge uv isin E(G)

such that kuv kge 0 then

WP(G) M2(G) minus (k + 1)M1(G) +(k + 1)2|E(G)|

minus 5Np(G) minus 3Nh(G)(5)

3 Numerical Computation

In this section we compute the numerical quantities forWiener polarity indices of lenvatinib and cabozantinibcancer drugs and chemical materials based on the silicateSierpinski and octahedral frameworks

31 Cancer Treatment Drugs Lenvatinib and cabozantinib[30] are orally accessible multikinase inhibitor and anti-neoplastic specialist that are utilized in the treatment ofadvanced metastatic medullary thyroid cancer (MTC) andhepatocellular carcinoma (HCC) Lenvatinib is a member ofthe class of quinolines that is the carboxamide of 4-3-chloro-4-[(cyclopropylcarbamoyl)amino]phenoxy-7-methoxyquinoline-6-carboxylic acid Cabozantinib is a di-carboxylic acid diamide that is N-phenyl-Nrsquo-(4-fluo-rophenyl)cyclopropane-11-dicarboxamide in which thehydrogen at position 4 on the phenyl ring is substituted byan (67-dimethoxyquinolin-4-yl)oxy group +e moleculargraph structures of lenvatinib and cabozantinib are shown inFigure 2

Theorem 2 Let G1 and G2 be the graphs of lenvatinib andcabozantinib 3en WP(G1) 48 and WP(G2) 62

Proof Since lenvatinib and cabozantinib are square-freegraphs we complete the proof by using +eorem 1 andTable 1 Let Eij uv isin E(G1) du i dv j1113864 1113865 such thatE(G1) cup 1leiltjle3Eij


WP G1( 1113857 1113944uvisinEij


1113944uvisinEij

1leiltjle3kuv0


1leiltjle3kuv1


48

(6)

H2N

O

N

O

ONH

HN

O

CH3

Cl

(a)

CH3

O

O

CH3

N

O

NH

OO

NH F

(b)




Nh(1 3) (2 2) (2 3) (3 3) (1 2) (3 4) (2 4)


w1

w2

x1 y1

y2

y3

x2

x3

u v

wkuv

xpndash1yqndash1

xp yq















2916n2

minus 3456n + 1242 minus 3 1134n2

minus 1242n + 4321113872 1113873 + 9 108n2

minus 108n + 361113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

459n2

minus 657n + 249


162n2


+ 99n minus 181113872 1113873 + 9 6n2

+ 12n1113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

12

51n2

+ 3n minus 361113872 1113873


864n2

minus 1056n + 384 minus 3 432n2

minus 480n + 1681113872 1113873 + 9 54n2


2

189n2

minus 267n + 99


48n2


+ 36n minus 81113872 1113873 + 9 3n2

+ 6n1113872 1113873 minus3n(n minus 1)

2

12

21n2

+ 3n minus 161113872 1113873

(7)







(a) (b)


(a) (b)





i dv j

WP Sn( 1113857 1113944

uvisinE Sn( )







43n

minus 32


1113872 1113873


(8)

(a) (b)


(a) (b)




G E(G) M1(G) M2(G)




DSL(n)


(a) (b)


(a) (b)






(a) (b)


(a) (b)





(2 3) 1 6


2 3nminus 1

(2 4) 1 6





(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References












































WP G1( 1113857 1113944uvisinEij


1113944uvisinEij

1leiltjle3kuv0


1leiltjle3kuv1


48

(6)

H2N

O

N

O

ONH

HN

O

CH3

Cl

(a)

CH3

O

O

CH3

N

O

NH

OO

NH F

(b)




Nh(1 3) (2 2) (2 3) (3 3) (1 2) (3 4) (2 4)


w1

w2

x1 y1

y2

y3

x2

x3

u v

wkuv

xpndash1yqndash1

xp yq















2916n2

minus 3456n + 1242 minus 3 1134n2

minus 1242n + 4321113872 1113873 + 9 108n2

minus 108n + 361113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

459n2

minus 657n + 249


162n2


+ 99n minus 181113872 1113873 + 9 6n2

+ 12n1113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

12

51n2

+ 3n minus 361113872 1113873


864n2

minus 1056n + 384 minus 3 432n2

minus 480n + 1681113872 1113873 + 9 54n2


2

189n2

minus 267n + 99


48n2


+ 36n minus 81113872 1113873 + 9 3n2

+ 6n1113872 1113873 minus3n(n minus 1)

2

12

21n2

+ 3n minus 161113872 1113873

(7)







(a) (b)


(a) (b)





i dv j

WP Sn( 1113857 1113944

uvisinE Sn( )







43n

minus 32


1113872 1113873


(8)

(a) (b)


(a) (b)




G E(G) M1(G) M2(G)




DSL(n)


(a) (b)


(a) (b)






(a) (b)


(a) (b)





(2 3) 1 6


2 3nminus 1

(2 4) 1 6





(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References
























































2916n2

minus 3456n + 1242 minus 3 1134n2

minus 1242n + 4321113872 1113873 + 9 108n2

minus 108n + 361113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

459n2

minus 657n + 249


162n2


+ 99n minus 181113872 1113873 + 9 6n2

+ 12n1113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

12

51n2

+ 3n minus 361113872 1113873


864n2

minus 1056n + 384 minus 3 432n2

minus 480n + 1681113872 1113873 + 9 54n2


2

189n2

minus 267n + 99


48n2


+ 36n minus 81113872 1113873 + 9 3n2

+ 6n1113872 1113873 minus3n(n minus 1)

2

12

21n2

+ 3n minus 161113872 1113873

(7)







(a) (b)


(a) (b)





i dv j

WP Sn( 1113857 1113944

uvisinE Sn( )







43n

minus 32


1113872 1113873


(8)

(a) (b)


(a) (b)




G E(G) M1(G) M2(G)




DSL(n)


(a) (b)


(a) (b)






(a) (b)


(a) (b)





(2 3) 1 6


2 3nminus 1

(2 4) 1 6





(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References















































(a) (b)


(a) (b)





i dv j

WP Sn( 1113857 1113944

uvisinE Sn( )







43n

minus 32


1113872 1113873


(8)

(a) (b)


(a) (b)




G E(G) M1(G) M2(G)




DSL(n)


(a) (b)


(a) (b)






(a) (b)


(a) (b)





(2 3) 1 6


2 3nminus 1

(2 4) 1 6





(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References














































i dv j

WP Sn( 1113857 1113944

uvisinE Sn( )







43n

minus 32


1113872 1113873


(8)

(a) (b)


(a) (b)




G E(G) M1(G) M2(G)




DSL(n)


(a) (b)


(a) (b)






(a) (b)


(a) (b)





(2 3) 1 6


2 3nminus 1

(2 4) 1 6





(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References













































G E(G) M1(G) M2(G)




DSL(n)


(a) (b)


(a) (b)






(a) (b)


(a) (b)





(2 3) 1 6


2 3nminus 1

(2 4) 1 6





(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References















































(a) (b)


(a) (b)





(2 3) 1 6


2 3nminus 1

(2 4) 1 6





(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References















































(a) (b)


(a) (b)



0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References












































0

2

4

6

8

10

12

14

16

18

WP

3 4 5 6 7 8 9 102n


RTOX (n)SnSTn

OT (n)DOT (n)

times104


1086m

4200

10n

86m

4210

n

05

1

15

2

020

times104

WP


(a) (b)




DOT(n)


(4 8) 108n2 minus 108n + 36





(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References















































(2) WP(DOT(n)) 1917n2 minus 2475n + 915






1113872 1113873 minus 3 3n2

minus 3n + 11113872 1113873

639n2

minus 279n minus 3


7776n2

minus 8928n + 3168 minus 3 2592n2

minus 2784n + 9601113872 1113873 + 9 216n2

minus 216n + 721113872 1113873 minus 3 9n2

minus 15n + 71113872 1113873

1917n2

minus 2475n + 915

(9)





32 otherwise



93n + 58) otherwise




du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(10)




2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References














































2) +en


du + dv( 1113857 + 1113944uvisinE48

du + dv( 1113857 + 1113944uvisinE88

du + dv( 1113857












12


(11)


4 Conclusion


Data Availability




References












































Wiener Polarity Index Calculation of Square-Free Graphs ...

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