# Topological Coindices and Quantitative Structure-Property Analysis of Antiviral Drugs Investigated in the Treatment of COVID-19

Authors: Syed Ajaz K. Kirmani; Parvez Ali; Jawed Ahmad
Journal: Journal of Chemistry
 (2022)

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## Abstract

SARS-CoV-2 is a new strain of coronavirus family that has never been previously detected in humans. This has grown into a huge public health issue that has affected people all around the world. Presently, there is no specific antiviral treatment for COVID-19. To tackle the outbreak, a number of drugs are being explored or have been utilized based on past experience. A molecular descriptor (or topological index) is a numerical value that describes a compound’s molecular structure and has been successfully employed in many QSPR/QSAR investigations to represent several physicochemical attributes. In order to determine topological characteristics of graphs, coindices (topological) take nonadjacent pair of vertices into account. In this study, we introduced CoM-polynomial and numerous degree-based topological coindices for several antiviral medicines such as lopinavir, ritonavir remdesivir, hydroxychloroquine, chloroquine, theaflavin, thalidomide, and arbidol which were studied using the CoM-polynomial approach. In the QSPR model, the linear regression approach is used to analyze the relationships between physicochemical properties and topological coindices. The findings show that the topological coindices under investigation have a substantial relationship with the physicochemical properties of possible antiviral medicines in question. As a result, topological coindices may be effective tools for studying antiviral drugs in the future for QSPR analyses.


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## Body

## 1. Introduction

COVID-19 is a disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which is a positive single-stranded RNA virus containing proteins that belong to the beta-coronavirus family. Beta-coronavirus can change its tissue tropism, host selection, and toxicity dramatically [1–3]. Some of the symptoms observed in patients who have been infected with SARS-CoV-2 include fever, cough, sore throat, rhinorrhea, severe pneumonia, bilateral lung opacities, and septic shock [4].COVID-19 is still untreatable with effective antiviral treatments despite the availability of several vaccinations to lower its severity. However, in order to combat the outbreak, many drugs are being evaluated or employed on prior studies. Lopinavir, ritonavir, chloroquine, hydroxychloroquine, azithromycin, remdesivir, arbidol, favipiravir, theaflavin, thalidomide, ribavirin, and others are among the medications used to combat the pandemic [5]. However, there is no consensus on their efficacy in treating COVID-19 infection. Many investigations are being conducted on these medications and their derivatives in order to employ them in the treatment of COVID-19 disease and to develop novel COVID-19 disease drugs [6–10].The process of drug discovery is multifaceted, costly, and time-consuming, and the use of computer-based approaches for design, synthesis, and biological screening of compounds provides an alternative to the old hit-or-miss strategy [11, 12]. Particularly, quantitative structure-property and activity relationship (QSPR/QSAR) modeling links a molecule’s physicochemical properties to its molecular structure.A descriptor provides numerical value to illustrate a compound’s molecular structure and molecular features. Although there are many other types of descriptors, topological descriptors, also known as topological indices, are frequently used in development of new drugs as they are data-rich, rapid to calculate, and have strong inclination towards prediction. The topological index, a graph invariant, is a numerical descriptor that encodes the topology of molecules represented as molecular graph [13, 14].Chemical compounds can be converted into molecular graphs by turning them into graphs with atoms as vertices and bonds as edges. In a molecular graph,G=V,E, VG, and EG are vertex and edge set, while VG and EG represent the number of vertices and edges in G. A degree of vertex is defined as the number of vertices that are adjacent to u∈VG and is denoted by du. The simple graph with the same vertex set VG is called complement of a graph G, denoted by G¯, such that any two vertices uv∈EG¯ if and only if uv∉EG [15].Wiener was the first to apply topological indexes in QSPR, demonstrating that his index, dubbed the Wiener index after him, was closely aligned with alkane boiling points [16]. The Wiener index has been used to describe a range of chemical and physical properties of molecules, as well as to associate a molecule's structure with its biological activity [17]. Since 1947, there have been hundreds of topological indexes constructed nonetheless, in chemical and mathematical literature, the Wiener index, Randic index, Zagreb indices, and its variant, are the most extensively utilized [16, 18–28]. Symmetric division index (SDD), inverse sum index, F-index, harmonic index, augmented Zagreb index, and redefined versions of Zagreb indices are among the indices regularly encountered in the literature [23, 29–34].A majority of degree-based topological indices take adjacent vertex pairs into consideration. However, over time, researchers have begun to incorporate nonadjacent pairing of vertices into consideration when computing some topological features of graphs, resulting in degree-based topological indices known as coindices.To enhance our ability to quantify the contributions of nonadjacent vertices to various aspects of molecules, Doslic [35] formalized the first and second Zagreb coindices of graph G as follows:(1)M¯1G=∑uv∉EGdu+dv,M¯2G=∑uv∉EGdudv.It is worth noting that Zagreb coindices ofG are not the same as Zagreb indices of G¯. The degrees are with respect to G, whereas the defining sums are over EG¯.Xu et al. [36] defined the multiplicative versions of Zagreb coindices described as follows:(2)∏1G¯=∏uv∉EGdu+dv,∏2G¯=∏uv∉EGdudv.De et al. [37] demonstrated that the logarithm of the octanol-water partition coefficient (log(P)) and the associated F-coindex values of octane isomers have a strong relationship (r = 0.96), leading to the introducing the F-coindex F¯ of G as follows:(3)F¯G=∑uv∉EGd2u+d2v.Recently Berhe et al. [38] computed various coindices of C4C8(S) nanotubes, nanotorus and graphene sheets. For more literature on the topological coindices, readers might refer to [39–44].Other degree-based topological coindices mentioned in this article can be defined in the same way as their classical degree counterparts:Second modified Zagreb coindex:mM¯2G=∑uv∉EG1/dudvRedefined third Zagreb coindex:ReZG3¯G=∑uv∉EGdudvdu+dvRandić coindex:R¯kG=∑uv∉EGdudvkInverse Randić coindex:RR¯kG=∑uv∉EG1/dudvkSymmetric division coindex:SDD¯G=∑uv∉EGd2u+d2v/dudvHarmonic coindex:H¯G=∑uv∉EG2/du+dvInverse sum indeg coindex:I¯G=∑uv∉EGdudv/du+dvAugmented Zagreb coindex:A¯G=∑uv∉EGdudv/du+dv−23In the literature, many distance-based graph polynomials have been proposed to accelerate the computation of certain graph indices [45–48]. Similarly, Deutsch and Klavzar [49] proposed the M-polynomial, a degree-based polynomial, which can be used to construct a number of indices. Because of its broad adaptability, it has been utilized in a number of articles to obtain topological indices [50–57]. Mondal et al. [58] recently examined four antiviral medicines for COVID-19 patients: remdesivir, chloroquine, hydroxychloroquine, and theaflavin. Kirmani et al. [59] studied eight antiviral drugs including lopinavir, ritonavir, arbidol, and thalidomide via M-polynomial and NM-polynomials and performed QSPR and QSAR to predict the strength of these topological indices. For recent topological results and QSPR/QSAR on drugs being used in COVID-19 treatment, refer [60–63]. Very recently, Yang et al. [64] consider nonadjacent vertices of molecular graph of COVID-19 drugs to investigate various topological coindices such as F-coindex, first Zagreb coindex, first multiplicative Zagreb coindex, second Zagreb coindex, and second multiplicative Zagreb coindex of hydroxyethyl starch conjugated with hydroxychloroquine used for COVID-19 treatment.As the M-polynomial takes into account the contributions of pairs of adjacent vertices, a polynomial that takes into account nonadjacent pairs of vertices is required which plays parallel role “to what M-polynomial does” to compute various types of graph coindices. This encourages us to cultivate polynomials based on nonadjacent pairs of vertices of chemical compounds. Therefore, in this article, we continue our investigation of antiviral drugs used in COVID-19 patients and define a CoM-polynomial by considering equivalent contributions from pairs of nonadjacent vertices, which capture and quantify a potential influence of distant pairs of vertices on the molecule's attributes. The true strength of the CoM-polynomial is its comprehensiveness, which includes useful information on degree-based graph invariants. Many degree-based topological coindices that link chemical features of the material under research are produced in closed forms by the CoM-polynomial using elementary calculus.Some physicochemical features of the drugs are used to test the predictive strength of these coindices. Using the linear regression models, we see that these coindices are found to have high correlation with the physicochemical properties except for polar surface area and surface tension. Additionally, Figure1 shows the surface representation of CoM-polynomials for these drugs that shows different behaviors by varying the parameters x and y. Figure 2 depicts the chemical structure of these chemical compounds.Figure 1
Plot of CoM-polynomials for various antiviral drugs.Figure 2
Antiviral drugs and their molecular structure.


## 2. Preliminaries

Let us use the following notation for the rest of paperni=Vi for Vi=v∈VG|dv=imij=Eij for Eij=uv∈EG|du=iandpdv=jm¯ij=E¯ij for E¯ij=uv∈EG¯|du=ianddv=jWe extend the concept of M-polynomial for nonadjacent pair of vertices and define CoM-polynomial as follows:(4)CoMG;x,y=M¯G;x,y=∑i≤jm¯ijGxiyj,where m¯ijG,i,j≥1,be the number of edgesuv∉EGsuch thatdu,dv=i,j.Various topological coindices derived from CoM-polynomial are listed in Table1.Table 1
Topological coindices and its derivation from CoM-polynomial.
Topological coindexFormulaϕdu,dvDerivation fromfx,y=CoMG,x,yFirst Zagreb coindex:M¯1G∑uv∉EGdu+dvDx+Dyfx,yx=y=1Second Zagreb coindex:M¯2G∑uv∉EGdudvDxDyfx,yx=y=1Second modified Zagreb coindex:mM¯2G∑uv∉EG1/dudvSxSyfx,yx=y=1Redefined third Zagreb coindex:ReZG3¯G∑uv∉EGdudvdu+dvDxDyDx+Dyfx,yx=y=1Forgotten topological coindex:F¯G∑uv∉EGd2u+d2vDx2+Dy2fx,yx=y=1Randić coindex:R¯kG∑uv∉EGdudvkDxkDykfx,yx=y=1Inverse Randić coindex:RR¯kG∑uv∉EG1/dudvkSxkSykfx,yx=y=1Symmetric division coindex:SDD¯ (G)∑uv∉EGχ2u+χ2v/dudvDxSy+SxDyfx,yx=y=1Harmonic coindex:H¯G∑uv∉EG2/du+dv2SxJfx,yx=1Inverse sum indeg coindex:I¯G∑uv∉EGdudv/du+dvSxJDxDyfx,yx=1Augmented Zagreb coindex:A¯G∑uv∉EGdudv/du+dv−23Sx3Q−2JDx3Dy3fx,yx=1WhereDx=x∂fx,y/∂x,Dy=y∂fx,y/∂y,Sx=∫0xft,y/tdt,Sy=∫0yfx,t/tdt,Jfx,y=fx,x,Qkfx,y=xkfx,yThe following lemma is due to Berhe [38].Lemma 1.
For a connected graphG of order n, we have(5)m¯ij=E¯ij=nini−12−miifori=jninj−mijfori<j.where ni, mij, and m¯ij are defined above.


## 3. Discussion and Main Results

This section uses combinatorial computation, edge partition technique, vertex partition technique, and nonedge counting method to obtain CoM-polynomials of molecular graphs of lopinavir (LOP), ritonavir (RIT), arbidol (ARD), thalidomide (THD), remdesivir (REM), Chloroquine (CLO), hydroxychloroqine (HYC), and theaflavin (THF). We also deduce many well-known topological coindices for the given molecular graphs.The expression of the CoM-polynomial for lopinavir is established first.Theorem 1.
The CoM-polynomial for LOP is given by(6)CoMLOP;x,y=104xy3+262x2y3+316x2y2+84x3y3.Proof.
From the Figure3, it is easy to conclude that VLOP=46 and ELOP=49. Also, the edge set of LOP may be categorized into five categories based on the degree of vertices:(7)E13=uv∈ELOP|du=1,dv=3,E23=uv∈ELOP|du=2,dv=3,E22=uv∈ELOP|du=2,dv=2,E33=uv∈ELOP|du=3,dv=3,such that m13=E13=8,E22=14,E23=20,andE33=7.
Similarly,VLOP can also be split into three classes based on the degree they have.(8)n1=V1=8,n2=V2=24,n3=V3=14.
Using Lemma1, we have(9)m¯13=n1n3−m13=8∗14−8=104,m¯22=n2n2−12−m22=24∗232−14=262.
Similarly,m¯23=E¯23=316 and m¯33=E¯33=84
By definition of CoM-polynomial,(10)CoMG;x,y=M¯G;x,y=∑i≤jm¯ijGxiyj,where m¯ijG,i,j≥1,be the number of edgesuv∉EGsuch thatdu,dv=i,j.(11)CoMLOP;x,y=∑1≤3m¯13xy3+∑2≤2m¯22x2y2+∑2≤3m¯23x2y3+∑3≤3m¯33x3y3=104xy3+262x2y2+316x2y3+84x3y3.
Hence, the result.
Using Theorem1 and Table 1, we recover some degree-based topological coindices (DBTCI) of the LOP in the following proposition.Figure 3
Molecular graph of lopinavir.Proposition 1.
The topological coindices for LOP are given byM¯1LOP=3548M¯2LOP=4012mM¯2LOP=162.1667ReZG3¯LOP=19456F¯LOP=8756R¯kLOP=104.3k+262.22k+316.2k3k+84.32kRR¯kLOP=104/3k+262/22k+316/2k3k+84/32kSDD¯LOP=1723.33H¯LOP=337.4I¯LOP=845.2A¯LOP=5931.8125Proof.
Let,(12)fx,y=CoMLOP;x,y=104xy3+262x2y3+316x2y2+84x3y3.
Then,(13)Dxfx,y=104xy3+524x2y2+632x2y3+252x3y3,Dyfx,y=312xy3+524x2y2+948x2y3+252x3y3,Dx+Dyfx,y=416xy3+1048x2y2+1580x2y3+504x3y3,DyDxfx,y=312xy3+1048x2y2+1896x2y3+756x3y3,(14)Dx2+Dy2fx,y=1040xy3+2096x2y2+4108x2y3+1512x3y3,DxkDykfx,y=3k104xy3+22k262x2y2+2k3k316x2y3+32k84x3y3,DxDyDx+Dyfx,y=1248xy3+4192x2y2+9480x2y3+4536x3y3,(15)SxSyfx,y=104xy33+131x2y22+158x2y33+28x3y33,SxkSykfx,y=104xy33k+262x2y222k+316x2y32k3k+84x3y332k,SyDx+SxDyfx,y=10403xy3+524x2y2+20543x2y3+168x3y3,SxJfx,y=366x44+316x55+84x66,SxJDyDxfx,y=340x4+1896x55+126x6,Sx3Q−2JDx3Dy3fx,y=2447x2+2528x3+93.2116x4.
With the help of Table1, we get the following:M¯1LOP=Dx+Dyfx,y|x=y=1=3548M¯2LOP=DxDyfx,y|x=y=1=4012mM¯2LOP==SxSyfx,y|x=y=1=162.1667ReZG3¯LOP=DxDyDx+Dyfx,y|x=y=1=19456F¯LOP=Dx2+Dy2fx,y|x=y=1=8756R¯kLOP=DxkDykfx,y|x=y=1==104.3k+262.22k+316.2k3k+84.32kRR¯kLOP=SxkSykfx,y|x=y=1=104/3k+262/22k+316/2k3k+84/32kSDD¯LOP=SyDx+SxDyfx,y|x=y=1=1723.33H¯LOP=2SxJfx,y|x=y=1=337.4I¯LOP=SxJDxDyfx,y|x=y=1=845.2A¯LOP=Sx3Q−2JDx3Dy3fx,y|x=y=1=5931.8125
The CoM-polynomial for Ritonavir’s molecular graph (RIT) is obtained in the following theorem.Theorem 2.
The CoM-polynomial forRIT is given by(16)CoMRIT;x,y=126xy3+312x2y2+364x2y3+100x3y3.Proof.
We haveVRIT=50 and ERIT=53 from the graph shown in Figure 4.
Partition of theERIT can be calculated based on the degree of vertices as follows:(17)E13=uv∈ER|du=1,dv=3,E23=uv∈ER|du=2,dv=3,E22=uv∈ER|du=2,dv=2,E33=uv∈ER|du=3,dv=3.such that m13=E13=9,m22=E22=13,m23=E23=26,m33=E33=5.
Likewise,VRIT can be classified into three groups based on their degrees.(18)n1=V1=9,n2=V2=26,n3=V3=15.
Using Lemma1, we have(19)m¯13=n1n3−m13=9∗15−9=126,m¯22=n2n2−12−m22=26∗252−13=312.
Similarly,m¯23=E¯23=364 and m¯33=E¯33=100
We have CoM-polynomial according to the definition:(20)CoMRIT;x,y=∑1≤3m¯13xy3+∑2≤2m¯22x2y2+∑2≤3m¯23x2y3+∑3≤3m¯33x3y3=126xy3+312x2y2+364x2y3+100x3y3.
Hence, the theorem.
Now, using the same procedure as in Proposition1, we may extract several topological coindices for RIT in the following proposition.Figure 4
Molecular graph of Ritonavir.Proposition 2.
The topological coindices forRIT are given byM¯1RIT=4172M¯2RIT=4170mM¯2RIT=191.778ReZG3¯RIT=22824F¯RIT=10288R¯kRIT=126..3k+312.22k+364.2k3k+100.32kRR¯kRIT=126/3k+312/22k+364/2k3k+100/32kSDD¯RIT=2032.667H¯RIT=397.933I¯RIT=993.3A¯RIT=6972.313
In the following theorem, expression for CoM-polynomal of Arbidol (ARD) is derived.Theorem 3.
The CoM-polynomial forARD is given by(21)CoMARD;x,y=76xy2+71xy3+49x2y2+112x2y3+46x3y3.Proof.
We haveVARD=29 and EARD=31 from Figure 5.
EARD and VARD can be grouped in following way:(22)m12=E12=1,m13=E13=6,m22=E22=6,m23=E23=9,m33=E33=9,n1=V1=7,n2=V2=11,n3=V3=11.
From Lemma1, we get(23)m¯12=E¯12=76,m¯13=E¯13=71,m¯22=E¯22=49,m¯23=E¯23=112,m¯23=E¯33=46.
Now,CoMARD;x,y can easily be obtained:(24)CoMARD;x,y=76xy2+71xy3+49x2y2+112x2y3+46x3y3.
The following preposition may be computed instantaneously using the aforementioned theorem.Figure 5
Molecular graph of arbidol.Proposition 3.
The topological coindices forARD are calculated asM¯1ARD=1544M¯2ARD=1647mM¯2ARD=97.694ReZG3¯ARD=7936F¯ARD=3766R¯kARD=76.2k+71.3k+49.22k+112.2k3k+46.32kRR¯kARD=76/2k+71/3k+49/22k+112/2k3k+46/32kSDD¯ARD=859.33H¯ARD=85.4I¯ARD=356.317A¯ARD= 2659.594The following theorem calculates the CoM-polynomial of the molecular graph (THD) of thalidomide.Theorem 4.
The CoM-polynomial forTHD is given by(25)CoMTHD;x,y=28xy3+17x2y2+50x2y3+21x3y3.Proof.
Figure6 depicts shows the molecular graph (THD) of thalidomide
We haveVTHD=19, ETHD=21 such that E13=4,E22=4,E23=6,E33=7 and(26)n1=V1=4,n2=V2=7,n3=V3=8.
Lemma1 gives(27)m¯13=E¯13=28,m¯22=E¯22=17,m¯23=E¯23=50,m¯33=E¯33=21.
Thus,CoMTHD;x,y can be calculated as(28)CoMTHD;x,y=28xy3+17x2y2+50x2y3+21x3y3.
This completes the proof.
In the light of Theorem4, the following results are yielded.Figure 6
Molecular graph of thalidomide.Proposition 4.
The topological coindices forTHD are calculated asM¯1THD=556M¯2THD=641mM¯2THD=24.25ReZG3¯THD=3242F¯THD=1444R¯kTHD=28.3k+17.22k+50.2k3k+21.32kRR¯kTHD=28/3k+17/22k+50/2k3k+21/32kSDD¯THD=277.667H¯THD=49.5I¯THD=129.5A¯THD=782.203Theorem 5.
The CoM-polynomial forREM is given by(29)CoMREM;x,y=177xy2+94xy3+17xy4+181x2y2+206x2y3+35x2y4+49x3y3+20x3y4.Proof.
By analyzing the structure of the REM shown in Figure7, we have(30)m12=E12=2,m13=E13=5,m14=E14=2,m22=E22=9,m23=E23=14,m24=E24=4,m33=E33=6,m34=E34=2.
Similarly for vertex set of REM
n1=V1=9, n2=V2=20, n3=V3=11, and n4=V4=2.
Also, Lemma1 gives(31)m¯12=E¯12=177,m¯13=E¯13=94,m¯14=E¯14=17,m¯22=E¯22=181,m¯23=E¯23=206,m¯24=E¯24=35,m¯33=E¯33=49,m¯34=E¯34=20.
Using above information, we can easily obtain(32)CoMREM;x,y=177xy2+94xy3+17xy4+181x2y2+206x2y3+35x2y4+49x3y3+20x3y4.
Now, using the aforementioned result, we get the Proposition5.Figure 7
Molecular graph of remdesivir.Proposition 5.
The topological coindices forREM are given byM¯1REM=3390M¯2REM=3625mM¯2REM=215.153ReZG3¯REM=17612F¯REM=8322R¯kREM=177.2k+94.3k+17.4k+181.22k+206.2k3k+35.23k+49.32k+20.3k4kRR¯kREM=177/2k+94/3k+17/4k+181/22k+206/2k3k+35/23k+49/32k+20/3k4kSDD¯REM=1863.583H¯REM=378.414I¯REM=784.752A¯REM= 5984.167Theorem 6.
The CoM-polynomial forCLO is given by(33)CoMCLO;x,y=46xy2+22xy3+61x2y2+60x2y3+13x3y3.Proof.
By means of structure analysis of molecular graph CLO depicted in Figure8, we have m12=2,m13=2,m22=5,m23=12,andm33=2 such that ECLO=23 and n1=V1=4, n2=V2=12, and n3=V3=6 vertex set of CLO, i.e., VCLO=22.
Now, applying Lemma1, we have(34)m¯12=46,m¯13=22,m¯22=61,m¯23=60,m¯33=13.
With the help of above information and definition of CoM-polynomial, it is easy to deduce(35)CoMCLO;x,y=46xy2+22xy3+61x2y2+60x2y3+13x3y3.Figure 8
Molecular graph of chloroquine.Proposition 6.
The topological coindices forCLO are given byM¯1CLO=848M¯2CLO=879mM¯2CLO=57.028ReZG3¯CLO=4066F¯CLO=1952R¯kCLO=46.2k+22.3k+61.22k+60.2k3k+13.32kRR¯kCLO=46/2k+22/3k+61/22k+60/2k3k+13/32kSDD¯CLO=466.333H¯CLO=100.5I¯CLO=219.167A¯CLO=1558.328Theorem 7.
The CoM-polynomial for HYC is given by(36)CoMHYC;x,y=50xy2+22xy3+72x2y2+66x2y3+13x3y3.Proof.
By means of analyzing Figure9, we have m12=2,m13=2,m22=6,m23=12,andm33=2 such that EHYC=24, and for the vertex set of HYC, i.e., VHYC=23, we have following groups n1=V1=4, n2=V2=13, and n3=V3=6.
Using above information Lemma1 gives(37)m¯12=50,m¯13=22,m¯22=72,m¯23=66,m¯33=13.
With the help of above information and definition of CoM-polynomial, it is easy to deduce(38)CoMHYC;x,y=50xy2+22xy3+72x2y2+66x2y3+13x3y3.Figure 9
Molecular graph of hydroxyhloroquine.Proposition 7.
The topological coindices for HYC are given byM¯1HYC=634M¯2HYC=967mM¯2HYC=62.778ReZG3¯HYC=2598F¯HYC=2138R¯kHYC=50.2k+22.3k+72.22k+66.2k3k+13.32kRR¯kHYC=50/2k+22/3k+72/22k+66/2k3k+13/32kSDD¯HYC=511.333H¯HYC=111.067I¯HYC=220.533A¯HYC=1726.328Theorem 8.
The CoM-polynomial forTHF is given by(39)CoMTHF;x,y=190xy3+198x2y3+176x3y3.Proof.
Let THF be the molecular graph of theaflavin (see Figure10) having ETHF=46 such that m13=10,m23=22,andm33=14. Its vertex VTHF=41 can be grouped into three classes on the basis of their degrees n1=V1=10, n2=V2=11, and n3=V3=20.
Now, applying Lemma1, we get(40)m¯13=190,m¯23=198,m¯33=176.
Next, it is easy to plug these values in definition of CoM-polynomial to obtain(41)CoMTHF;x,y=190xy3+198x2y3+176x3y3.Figure 10
Molecular graph of theaflavin.Proposition 8.
The topological coindices forTHF are given byM¯1THF=2806M¯2THF=3342mM¯2THF=115.889ReZG3¯THF=17724F¯THF=7642R¯kTHF=190.3k+198.2k3k+176.32kRR¯kTHF=190/3k+198/2k3k+176/32kSDD¯THF=1414.333H¯THF=232.867I¯THF=644.1A¯THF=4230


## 4. Quantitative Structure Analysis via Topological Coindices

The aim of this piece is to assess the efficiency of topological coindices and to develop a QSPR between the various topological coindices stated above and specific physicochemical aspects of the drugs being studied for COVID-19 treatment. For this, the benchmark set eight of physicochemical properties of these drugs are taken from ChemSpider (http://www.chemspider.com) and presented in Table 2. However, some physicochemical properties like boiling point, enthalpy of vaporization, and flash point for remdesivir is not available in literature.Table 2
COVID-19 drugs and their physicochemical properties.
DrugsBoiling point (BP)Enthalpy of vaporization: (E)Flash point: (FP)Molar refractivity (MR)Polar surface area (PSA)Polarizability (P)Surface tension (T)Molar volume (MV)Lopinavir924.2140.8512.7179.212071.049.5540.5Ritonavir947.0144.4526.6198.920278.953.7581.7Arbidol591.891.5311.7121.98048.345.3347.3Thalidomide487.879.4248.865.28725.971.6161Chloroquine460.672.1232.397.42838.644.0287.9Hydroxychloroquine516.783.0266.399.04839.249.8285.4Theaflavin1003.9153.5336.5137.321854.4138.6301.0Remdesivir———149.521359.362.3409The values of the drugs’ various topological coindices are reported in Table3, while Table 4 contains the linear regression model’s correlation coefficient (r) between coindices and properties of various COVID-19 drugs.Table 3
The values of topological coindices for molecular graph of COVID-19 Drugs.
DrugsM¯1M¯2mM¯2ReZG3¯F¯SDD¯H¯I¯A¯Lopinavir35484012162.1671945687561723.333337.4845.25931.8123Ritonavir41724710191.77822824102882032.667397.933993.36972.313Arbidol1544164797.69479363766859.33385.4356.3172659.594Thalidomide55664124.2532421444277.66749.5129.5782.203Chloroquine84887957.02840661952466.333100.5219.1671558.328Hydroxychloroquine63496762.77825982138511.333111.067220.5331726.328Theaflavin28063342115.8891772476421414.333232.867644.14230Remdesivir33903625215.1531761283221863.583378.414784.7525984.167Table 4
The linear regression model's correlation coefficient (r) between topological coindices and physicochemical features of various medications for COVID-19 patients
CoindexBoiling point (BP)Enthalpy of vaporization (E)Flash point: (FP)Molar refractivity (MR)Polar surface area (PSA)Polarizability (P)Surface tension (T)Molar volume (MV)M¯10.9320.9240.9490.9550.8290.9560.1640.874M¯20.9480.9420.9430.9580.8340.9580.2040.869mM¯20.8610.8490.9540.8950.7750.896-0.0030.843ReZG3¯0.9630.9570.9140.9340.8610.9340.2740.828F¯0.9570.9510.9330.9490.8520.9490.2280.853SDD¯0.9330.9260.9460.9470.8390.9470.1480.865H¯0.8810.8760.9500.9120.7900.9120.0740.855I¯0.9220.9150.9590.9680.8070.9680.1310.896A¯0.9000.8920.9670.9640.7940.9650.0720.904Table4 demonstrates that I¯ has a strong positive association (r = 0.97) with molar refractivity (MR) and polarizability (P), whereas A¯ has the significant correlation (r = 0.97) with flash point (FP).Among the linear regression models, the most suitable coindices for modeling the various for physicochemical properties are as follows:(1)
TheReZG3¯ for BP, E, and PSA. However, ReZG3¯ does not predict PSA strongly.(2)
TheI¯ for MR and P.(3)
TheA¯ for FP and MV.Among the linear regression models, there is no topological coindices that can predict surface tension (T) well. The best compatible fitting and predictive linear regression models for ReZG3¯ (Co-ReZG3), I¯ (Co-I), and A¯ (Co-A) with physicochemical attributes of the drugs are plotted in Figure 11.Figure 11
Plot of the most compatible fitting and predictive linear regression models for the different coindices with physicochemical properties of the drugsr≥0.9.These findings suggest that topological coindices can predict the properties of the drugs in question. Therefore, topological coindices could be a useful tool for QSPR analysis in antiviral therapy.


## 5. Conclusion

The structural features of chemical molecules are critical for the development of innovative products in pharmaceutical sciences, and they can be discovered using molecular descriptors known as topological indices and topological coindices.The CoM-polynomial described in this article is capable of creating closed forms of a large number of degree-based topological coindices that correlate chemical properties of the substance under investigation. As a result, in this study, we examine a variety of prospective COVID-19 therapies and analyze their structure to determine the CoM-polynomial for these treatments. Then, these polynomials are also used to generate a number of degree-based topological coindices. Linear models are investigated using eight descriptors and eleven topological coindices in the QSPR study. The numerous topological coindices calculated have shown to have good prediction ability for the properties of these prospective COVID-19 therapies. In particular the best predictive topological coindices are as follows:I¯showing strong positive correlation value with MR and P and A¯ which predicts FP strongly among all correlations. Because the quantitative structure-activity relationship model (QSAR) was not included in this study, it could be used in future studies for antiviral medications that treat COVID-19.



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