# M-Polynomial and NM-Polynomial of Used Drugs against Monkeypox

Authors: Özge Çolakoğlu; Muhammad Kamran; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2022)

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## Abstract

Monkeypox has recently re-emerged. Since monkeypox has started to be seen in many countries without an epidemic, studies have started on new drugs used to treat this disease. Drug discovery is a challenging process. To alleviate this process, molecular descriptors used to predict the physical, chemical, and biological properties of chemical structures are studied. Molecular descriptors are numerical values of a chemical structure represented by a graph. In this study, M-polynomial and neighborhood M-polynomial molecular descriptors of drugs used and candidates for use in monkeypox are calculated and these descriptors are compared.


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## Body

## 1. Introduction

Monkeypox virus belongs to theOrthopoxvirus genus of the Poxviridae family. The disease was named monkeypox when it was first discovered in monkeys in 1958. In 1970, the first human case was seen a child in the Democratic Republic of the Congo. Monkeypox shows symptoms very similar to smallpox. Monkeypox is a viral zoonosis, that is, a virus transmitted from animals to humans. Symptoms of monkeypox include fever, swollen lymph nodes, blisters on the body, and crusty rashes on the body. People infected with this virus begin to show these symptoms between five and twenty-one days [1].The World Health Organization (WHO) said that “Since 13 May 2022, cases of monkeypox have been reported to WHO from three WHO regions and 12 Member States that are not endemic for monkeypox virus” [1].Although there is no specific vaccine or drug against monkeypox, historically, the smallpox vaccine has been shown to be protective against this disease. So, for the treatment of monkeypox, the MVA-BN vaccine in 2019 and the tecovirimat drug in 2022 were approved [1].After the recent COVID-19 epidemic, the importance of rapid drug and vaccine discoveries has emerged. Drug discovery is not only a time-consuming process but also a costly process. In order to shorten this process, models that can predict the physicochemical and bioactivity properties of the newly produced molecular structure are obtained with topological indices studied in chemical mathematics [2–6]. These models are obtained via the structure-property/activity relationship (QSPR/QSAR) studies using molecular descriptors [7, 8].Molecular descriptors are numerical descriptors of graphs obtained via chemical structures [9]. In 1947, Wiener estimated some properties of alkanes using the index [10]. Since then, many molecular descriptors have been studied to predict the properties of chemical structures without experimentation. Recently, M-polynomials and NM-polynomials have been defined and studied for many chemical structures (see [11–22]).


## 2. Preliminaries

IfG graph has p,q∈V vertex set and pq∈E edge set, then the order (size) of vertices (edges) is defined as V (E. The degree of the vertex p of G graph is denoted by dp. The neighborhood of a vertex p in a graph G, Np, is the set of all vertices adjacent to p [23]. Let τij=Ei,jdp,dp=i,j and δq=∑p∈Nqdp and τij∗=Ei,j´|δp=i,δq=j.M-polynomial of a G graph is defined in [19] as(1)MG=∑i≤jτijrisj.The neighborhoodM-polynomial is defined in [20] as(2)NMG=∑i≤jτij∗risj.Degree-based molecular descriptors and neighborhood degree-based molecular descriptors are, respectively, defined as(3)DTIG=∑pq∈Eφdp,dq,andNDTIG=∑pq∈Ei,j´φδp,δq.In this paper, the first Zagreb (M1), second Zagreb (M2), second modified Zagreb (mM2), forgotten (F), redefined third Zagreb (ReM3), symmetric division (SDD), inverse sum indeg ISI, harmonic (H), and augmented Zagreb (A) indices are degree-based molecular descriptors, and third version of Zagreb (M1´), neighborhood second modified Zagreb (M2´), neighborhood second modified Zagreb (mM2′), neighborhood forgotten (F′), third NDe (ND3), fifth NDe (ND5), neighborhood inverse sum indeg (ISI´), neighborhood harmonic (H′), and Sanskruti (S) indices are neighborhood degree-based descriptors.Table1 shows molecular descriptors based on degrees and the sum of adjacent vertex degrees.Table 1
Molecular descriptors (see [19, 20]).
Molecular descriptorsMathematical expressionsDerivation FromMG/NMG´M1G/M1′G∑uv∈EGμu+μvDr+Dsφr,sr=s=1M2G/M2′G∑uv∈EGμuμvDrDsφr,sr=s=1mM2/mM2′∑uv∈EG1/μuμvErEsφr,sr=s=1F/F′∑uv∈EGμu2+μu2Dr2+Ds2φr,sr=s=1ReM3/ND3∑uv∈EGμuμvμu+μvDrDsDr+Dsφr,sr=s=1SDD/ND5∑uv∈EGμu/μv+μv/μuDrEs+ErDsφr,sr=s=1ISI/ISI′∑uv∈EGμuμv/μu+μvErJDrDsφr,sr=1H/H′∑uv∈EG2/μu+μv2ExJφr,sr=1A/S∑uv∈EGμuμv/μu+μv−23Ex3Ψ−2JDx3Dy3φr,sr=1The following operators are used in Table1:(4)Dr=r∂φr,s∂r,Ds=s∂φr,s∂s,Er=∫0rφt,rtdt,Es=∫0sφr,ttdt,Jφr,s=φr,r,Ψkφr,s=rkφr,s.Ifμp=dp and μq=dq, then φr,s=MG. If μp=δp and μq=δq, then φr,s=NMG.The used drugs for monkeypox are as follows.Tecovirimat (also known as Tpoxx) is an antiviral medication. This drug is approved by the United States Food and Drug Administration (FDA) for the treatment of human smallpox disease. Moreover, it can be also used for the treatment of non-variola orthopoxviruses (including monkeypox) in an outbreak. Its chemical formula is C19H15F3N2O3. Figure1 shows the structure of tecovirimat [24].Figure 1
The structure of tecovirimat [24].Cidofovir (also known as Vistide) is an antiviral medication. It is approved by the FDA for the treatment of cytomegalovirus retinitis in patients with acquired immunodeficiency syndrome (AIDS). Moreover, it is allowed for the use for the treatment of orthopoxviruses (including monkeypox) in an outbreak. Its chemical formula is C8H14N3O6P. Figure 2 shows the structure of cidofovir [24].Figure 2
The structure of cidofovir [24].Brincidofovir (also known as Tembexa) (CMX001) is an antiviral medication that is a lipid conjugate of cidofovir. On June 4, 2021, it was approved by the FDA for the treatment of human smallpox disease in adult and paediatric patients, including neonates. This drug is currently developing for use in the treatment of monkeypox. Its chemical formula is C27H52N3O7P. Figure 3 shows the structure of brincidofovir [24].Figure 3
The structure of brincidofovir [24].In this study, the degree and neighborhood degree based on molecular descriptors of these molecules are computed, which will be useful in predicting the biological activity and physicochemical properties of new drugs that can be obtained with possible combinations of considered and used drugs for treatment of monkeypox.


## 3. TheM-Polynomials and NM-Polynomials of the Chemical Structures of Used and Potential Drugs against Monkeypox

TheM-polynomials and NM-polynomials of the graphs obtained from tecovirimat, cidofovir, and brincidofovir drugs are found. Using these polynomials, various molecular descriptors depending on the degree and the sum of neighborhood vertex degrees for these structures are calculated.Theorem 1.
IfG1is the chemical graph of tecovirimat, then(5)MG1,r,s=3rs3+3rs4+3r2s2+10r2s3+11r3s3+r3s4.Proof.
From Figure1, the chemical graph of tecovirimat has 27 orders and 31 sizes, and also we can partition it as E1,3=3, E1,4=3, E2,3=10, E2,2=3, E3,3=11, E3,4=1. From equation (1),(6)MG1,r,s=∑1≤3τ13rs3+∑1≤4τ14rs4+∑2≤2τ22r2s2+∑2≤3τ23r2s3+∑3≤3τ33r3s3+∑3≤4τ34r3s4,orMG1,r,s=3rs3+3rs4+3r2s2+10r2s3+11r3s3+r3s4.Corollary 1.
Degree based on some molecular descriptors of the graph of tecovirimat:(1)
M1G1=162.(2)
M2G1=204.(3)
mM2G1=197/36.(4)
FG1=458.(5)
RM3G1=1122.(6)
SDdegG1=149/2.(7)
ISIG1=5301/140.(8)
HG1=1276/105.(9)
AG1=260.35698.Proof.
From Table1 and Theorem 1, we have(7)DrMG1,r,s=3rs3+3rs4+6r2s2+20r2s3+33r3s3+3r3s4,DsMG1,r,s=9rs3+12rs4+6r2s2+30r2s3+33r3s3+4r3s4,DrDsMG1,r,s=9rs3+12rs4+12r2s2+60r2s3+99r3s3+12r3s4,ErEsMG1,r,s=rs3+34rs4+34r2s2+53r2s3+119r3s3+112r3s4,Dr2MG1,r,s=3rs3+3rs4+12r2s2+40r2s3+99r3s3+9r3s4,Dy2MG1,r,s=27rs3+48rs4+12r2s2+90r2s3+99r3s3+16r3s4,DrDsDr+DsMG1,r,s=36rs3+60rs4+48r2s2+300r2s3+594r3s3+147r3s4,DrEs+ErDsMG1,r,s=10rs3+514rs4+6r2s2+653r2s3+22r3s3+2512r3s4+9rs3,ErJDrDsMG1,r,s=94r4+125r5+3r4+12r5+332r6+127r7,2ErJMG1,r,s=3r4+165r5+113r6+27r7,Er3Ψ−2JDr3Ds3MG1,r,s=2738r2+7849r3+801964r4+1728125r5.
The following results are obtained:(1)
M1G1=12rs3+15rs4+12r2s2+50r2s3+66r3s3+7s41,1=162.(2)
M2G1=9rs3+12rs4+12r2s2+60r2s3+99r3s3+12r3s41,1=204.(3)
mM2G1=rs3+3/4rs4+3/4r2s2+5/3r2s3+11/9r3s3+1/12r3s41,1=197/36.(4)
FG1=30rs3+51rs4+24r2s2+130r2s3+198r3s3+25r3s41,1=458.(5)
RM3G1=36rs3+60rs4+48r2s2+300r2s3+594r3s3+147r3s41,1=1122.(6)
SDdegG1=10rs3+51/4rs4+6r2s2+65/3r2s3+22r3s3+25/12r3s4+9rs31,1=149/2.(7)
ISIG1=9/4r4+12/5r5+3r4+12r5+33/2r6+12/7r71=1387/30.(8)
HG1=3r4+16/5r5+11/3r6+2/7r71=5301/140.(9)
AG1=273/8r2+784/9r3+8019/64r4+1728/125r51=260.35698.Theorem 2.
The neighborhood M-polynomial ofG1is(8)NMG1,r,s=r3s6+2r3s7+3r4s6+3r5s5+2r5s7+4r5s8+r6s6+r6s7+4r6s8+2r7s8+2r7s9+3r8s8+2r8s9+r9s9.Proof.
From Figure1, NEˇ3,6=1, NEˇ3,7=2, NEˇ4,6=3, NEˇ5,5=3, NEˇ5,7=2, NEˇ5,8=4, NEˇ6,6=1, NEˇ6,7=1, NEˇ6,8=4, NEˇ7,8=2, NEˇ7,9=2, NEˇ8,8=3, NEˇ8,9=2, NEˇ9,9=1.
From equation (2),(9)NMG1,r,s=∑3≤6τ36∗r3s6+∑3≤7τ37∗r3s7+∑4≤6τ46∗r4s6+∑5≤5τ55∗r5s5+∑5≤7τ57∗r5s7+∑5≤8τ58∗r5s8+∑6≤6τ66∗r6s6+∑6≤7τ67∗r6s7+∑6≤8τ68∗r6s8+∑7≤8τ78∗r7s8+∑7≤9τ79∗r7s9+∑8≤8τ88∗r8s8+∑8≤9τ89∗r8s9+∑9≤9τ99∗r9s9,orNMG1,r,s=r3s6+2r3s7+3r4s6+3r5s5+2r5s7+4r5s8+r6s6+r6s7+4r6s8+2r7s8+2r7s9+3r8s8+2r8s9+r9s9.Corollary 2.
The molecular descriptors based on the neighborhood degree ofG1graph are(1)
M1´G1=408.(2)
M2´G1=1362.(3)
mM2′G1=0.714.(4)
F′G1=2850.(5)
NDe3G1=16448.(6)
ND5G1=331/5.(7)
ISI´G1=99.298.(8)
H′G1=4.900.(9)
SG1=75874.123.Proof.
Using Theorem3, we obtain the following equations:(10)DrNMG1,r,s=3r3s6+6r3s7+12r4s6+15r5s5+10r5s7+20r5s8+6r6s6+6r6s7+24r6s8+14r7s8+14r7s9+24r8s8+16r8s9+9r9s9,DsNMG1,r,s=6r3s6+14r3s7+18r4s6+15r5s5+14r5s7+32r5s8+6r6s6+7r6s7+32r6s8+16r7s8+18r7s9+24r8s8+18r8s9+9r9s9,DrDsNMG1,r,s=18r3s6+42r3s7+72r4s6+75r5s5+70r5s7+160r5s8+36r6s6+42r6s7+192r6s8+112r7s8+126r7s9+192r8s8+144r8s9+81r9s9,ErEsNMG1,r,s=118r3s6+221r3s7+18r4s6+325r5s5+235r5s7+110r5s8+136r6s6+142r6s7+112r6s8+124r7s8+263r7s9+364r8s8+136r8s9+181r9s9,Dr2NMG1,r,s=9r3s6+18r3s7+48r4s6+75r5s5+50r5s7+100r5s8+36r6s6+36r6s7+144r6s8+98r7s8+98r7s9+192r8s8+128r8s9+81r9s9,Ds2NMG1,r,s=36r3s6+98r3s7+108r4s6+75r5s5+98r5s7+256r5s8+36r6s6+49r6s7+256r6s8+128r7s8+162r7s9+192r8s8+162r8s9+81r9s9,DrDsDr+DsNMG1,r,s=162r3s6+420r3s7+720r4s6+750r5s5+840r5s7+2080r5s8+432r6s6+611r6s7+2688r6s8+1680r7s8+2016r7s9+3072r8s8+2448r8s9+1458r9s9,DrEs+ErDsNMG1,r,s=6r3s6+14r3s7+18r4s6+15r5s5+14r5s7+32r5s8+6r6s6+7r6s7+32r6s8+16r7s8+18r7s9+24r8s8+18r8s9+9r9s9ErJDrDsNMG1,r,s=94r4+125r5+3r4+12r5+332r6+127r7,2ErJNMG1,r,s=3r4+165r5+113r6+27r7,Er3Ψ−2JDr3Ds3NMG1,r,s=2738r2+7849r3+801964r4+1728125r5.
The following results are obtained:(1)
M1´G1=9r3s6+20r3s7+30r4s6+30r5s5+24r5s7+52r5s8+12r6s6+13r6s7+56r6s8+30r7s8+32r7s9+48r8s8+34r8s9+18r9s91,1=408.(2)
M2´G1=18r3s6+42r3s7+72r4s6+75r5s5+70r5s7+160r5s8+36r6s6+42r6s7+192r6s8+112r7s8+126r7s9+192r8s8+144r8s9+81r9s91,1=1362.(3)
mM2′G1=1/18r3s6+2/21r3s7+1/8r4s6+3/25r5s5+2/35r5s7+1/10r5s8+1/36r6s6+1/42r6s7+1/12r6s8+1/24r7s8+2/63r7s9+3/64r8s8+1/36r8s9+1/81r9s91,1=0.714.(4)
F′G1=45r3s6+116r3s7+156r4s6+150r5s5+148r5s73256r5s8+72r6s6+85r6s7+400r6s8+226r7s8+260r7s9+384r8s8+290r8s9+162r9s91,1=2850.(5)
NDe3G1=162r3s6+420r3s7+720r4s6+750r5s5+840r5s7+2080r5s8+432r6s6+611r6s7+2688r6s8+1680r7s8+2016r7s9+3072r8s8+2448r8s9+1458r9s91,1=16448.(6)
ND5G1=dxEy+ExdyNMG1,x,y1,1=331/5.(7)
ISI´G1=ExJdxdyNMG1,x,y1=99.2983.(8)
H′G1=2ExJNMG1,x,y1=4.900.(9)
SG1=Ex3Ψ−2Jdx3dy3NMG1,x,y1=75874.123.Theorem 3.
IfG2is the chemical graph of cidofovir, then(11)MG2,r,s=rs2+2rs3+3rs4+2r2s2+8r2s3+r2s4+r3s3.Proof.
From Figure2, the chemical graph of cidofovir has 18 sizes and 18 orders. We can partition it as E1,2=1, E1,3=2, E1,4=3, E2,2=2, E2,3=8, E2,4=1, E3,3=1. From equation (1),(12)MG2,r,s=∑1≤2τ12rs2+∑1≤3τ13rs3+∑1≤4τ14rs4+∑2≤2τ22r2s2+∑2≤3τ23r2s3+∑2≤4τ24r2s4+∑3≤3τ33r3s3,orMG2,r,s=rs2+2rs3+3rs4+2r2s2+8r2s3+r2s4+r3s3.Corollary 3.
Degree based on some molecular descriptors of the chemical graph of cidofovir:(1)
M1G2=86.(2)
M2G2=93.(3)
mM2G2=287/72.(4)
FG2=234.(5)
RM3G2=464.(6)
SDdegG2=59.(7)
ISIG2=17.(8)
HG2=116/15.(9)
AG2=121,251.Proof.
From Table1 and Theorem 3, the following equations are obtained:(13)M1G2=3rs2+8rs3+15rs4+8r2s2+40r2s3+6r2s4+6r3s31,1=86,M2G2=2rs2+6rs3+12rs4+8r2s2+48r2s3+8r2s4+9r3s31,1=93,mM2G2=12rs2+23rs3+34rs4+12r2s2+43r2s3+18r2s4+19r3s31,1=28772,FG2=5rs2+20rs3+51rs4+16r2s2+104r2s3+20r2s4+18r3s31,1=234,RM3G2=6rs2+24rs3+60rs4+32r2s2+240r2s3+48r2s4+54r3s31,1=464,SDdegG2=52rs2+203rs3+514rs4+4r2s2+523r2s3+52r2s4+2r3s31,1=59,ISIG2=23r3+72r4+5r5+176r61=17,HG2=23r3+2r4+225r5+23r61=11615,AG2=8r+18r2+6409r3+124164r41=121,251.Theorem 4.
The neighborhood M-polynomial ofG2is(14)NMG2,r,s=r2s4+r3s5+r3s6+3r4s5+r4s6+r5s5+4r5s6+r5s7+2r6s6+2r6s7.Proof.
From Figure2, the set of edges is divided as follows: NEˇ2,4=1, NEˇ3,5=1, NEˇ3,6=1, NEˇ4,5=3, NEˇ4,6=1, NEˇ5,5=2, NEˇ5,6=4, NEˇ5,7=1, NEˇ6,6=2, NEˇ6,7=2.
From (2),(15)NMG1,r,s=∑2≤4τ24∗r2s4+∑3≤5τ35∗r3s5+∑2≤4τ36∗r3s6+∑4≤5τ45∗r4s5+∑4≤6τ46∗r4s6+∑5≤5τ55∗r5s5+∑5≤7τ57∗r5s7+∑5≤7τ56∗r5s6+∑6≤6τ66∗r6s6+∑6≤7τ67∗r6s7,orNMG2,r,s=r2s4+r3s5+r3s6+3r4s5+r4s6+r5s5+4r5s6+r5s7+2r6s6+2r6s7.Corollary 4.
The molecular descriptors based on the neighborhood degree of theG1graph are(1)
M1′G2=196.(2)
M2′G2=486.(3)
mM2′G2=0.783.(4)
F′G2=1006.(5)
NDe3G2=5306.(6)
ND5G2=5303/140.(7)
ISI´G2=45.562.(8)
H′G2=3.607.(9)
SG2=594.295.Proof.
Using Theorem4, the following equations are obtained:(16)M1′G2=6r2s4+8r3s5+9r3s6+27r4s5+10r4s6+10r5s5+44r5s6+12r5s7+24r6s6+26r6s71,1=196,M2′G2=8r2s4+15r3s5+18r3s6+60r4s5+24r4s6+25r5s5+120r5s6+35r5s7+72r6s6+84r6s71,1=486,mM2′G2=18r2s4+115r3s5+118r3s6+320r4s5+124r4s6+125r5s5+215r5s6+135r5s7+136r6s6+121r6s71,1=0.783,F′G2=20r2s4+34r3s5+45r3s6+123r4s5+52r4s6+50r5s5+244r5s6+74r5s7+144r6s6+170r6s71,1=1006,NDe3G2=48r2s4+120r3s5+162r3s6+540r4s5+240r4s6+250r5s5+1320r5s6+420r5s7+864r6s6+1092r6s71,1=5306,ND5G2=52r2s4+3415r3s5+52r3s6+12320r4s5+136r4s6+2r5s5+12215r5s6+7435r5s7+4r6s6+8521r6s71,1=5303140,ISI´G2=43r6+158r8+263r9+4910r10+8011r11+8413r13+9r121=45.562,H′G2=13r6+14r8+89r9+25r10+811r11+12r12+413r131=3.607,SG1=8r4+1258r6+29832343r7+29449512r8+400027r9+1361871000r10+14817611r111=594.295.Theorem 5.
IfG3is the chemical graph of brincidofovir, then(17)MG3,r,s=2rs2+2rs3+2rs4+21r2s2+8r2s3+2r2s4+r3s3.Proof.
From Figure3, the chemical graph of brincidofovir has 38 sizes and 38 orders. We can partition it as E1,2=1, E1,3=2, E1,4=3, E2,2=2, E2,3=8, E2,4=1, E3,3=1. From equation (1),(18)MG3,r,s=∑1≤2τ12rs2+∑1≤3τ13rs3+∑1≤4τ14rs4+∑2≤2τ22r2s2+∑2≤3τ23r2s3+∑2≤4τ24r2s4+∑3≤3τ33r3s3,orMG3,r,s=2rs2+2rs3+2rs4+21r2s2+8r2s3+2r2s4+r3s3.Corollary 5.
Degree based on some molecular descriptors of the chemical graph of brincidofovir:(1)
M1G3=166.(2)
M2G3=175.(3)
mM2G3=82/9.(4)
FG3=394.(5)
RM3G3=802.(6)
SDdegG3=173/2.(7)
ISIG3=196/5.(8)
HG3=107/6.(9)
AG3=286.881.Proof.
From Table1 and Theorem 5, the following equations are obtained:(19)M1G3=6rs2+8rs3+10rs4+84s2+40r2s3+12r2s4+6r3s31,1=166,M2G3=4rs2+6rs3+8rs4+84r2s2+48r2s3+16r2s4+9r3s31,1=175,mM2G3=rs2+23rs3+12rs4+214r2s2+43r2s3+14r2s4+19r3s31,1=829,FG3=10rs2+20rs3+34rs4+168r2s2+104r2s3+40r2s4+18r3s31,1=394,RM3G3=12rs2+24rs3+40rs4+336r2s2+240r2s3+96r2s4+54r3s31,1=802,SDdegG3=5rs2+203rs3+172rs4+42r2s2+523r2s3+5r2s4+2r3s31,1=1732,ISIG3=43r3+452r4+565r5+256r61=1965,HG3=43r3+232r4+4r5+r61=1076,AG3=16r+6994r2+185627r3+175364r41=286.881.Theorem 6.
The neighborhood M-polynomial ofG3is(20)NMG3,r,s=r2s3+r2s4+r3s4+r3s5+r3s6+17r4s4+4r4s6+r5s5+3r5s6+r5s7+4r6s6+2r6s7.Proof.
From Figure3, the set of edges is divided as follows: NEˇ2,3=1, NEˇ2,4=1, NEˇ3,4=1, NEˇ3,5=1, NEˇ3,6=1, NEˇ4,4=17, NEˇ4,6=4, NEˇ5,5=2, NEˇ5,6=3, NEˇ5,7=1, NEˇ6,6=4, NEˇ6,7=2.
From equation (2),(21)NMG1,r,s=∑2≤3τ23∗r2s3+∑2≤4τ24∗r2s4+∑3≤4τ34∗r3s4+∑3≤5τ35∗r3s5+∑3≤6τ36∗r3s6+∑4≤4τ44∗r4s4+∑4≤6τ46∗r4s6+∑5≤5τ55∗r5s5+∑5≤7τ57∗r5s7+∑5≤6τ56∗r5s6+∑6≤6τ66∗r6s6+∑6≤7τ67∗r6s7,orNMG3,r,s=r2s3+r2s4+r3s4+r3s5+r3s6+17r4s4+4r4s6+r5s5+3r5s6+r5s7+4r6s6+2r6s7.Corollary 6.
Some molecular descriptors based on the neighborhood degree ofG3graph are(1)
M1′G3=350.(2)
M2′G3=830.(3)
mM2′G3=2.093.(4)
F′G3=1654.(5)
NDe3G3=8060.(6)
ND5G3=32947/420.(7)
ISI´G3=79.303.(8)
H′G3=8.627.(9)
SG3=1005.794.Proof.
Using Theorem6, the following equations are obtained:(22)M1′G3=5r2s3+6r2s4+7r3s4+8r3s5+9r3s6+136r4s4+40r4s6+10r5s5+33r5s6+12r5s7+48r6s6+26r6s71,1=350,M2′G3=6r2s3+8r2s4+12r3s4+15r3s5+18r3s6+272r4s4+96r4s6+25r5s5+90r5s6+35r5s7+144r6s6+84r6s71,1=830,mM2′G3=16r2s3+18r2s4+112r3s4+115r3s5+118r3s6+1716r4s4+16r4s6+125r5s5+110r5s6+135r5s7+19r6s6+121r6s71,1=2.093,F′G3=13r2s3+20r2s4+25r3s4+34r3s5+45r3s6+544r4s4+208r4s6+50r5s5+183r5s6+74r5s7+288r6s6+170r6s71,1=1654,NDe3G3=30r2s3+48r2s4+84r3s4+120r3s5+162r3s6+2176r4s4+960r4s6+250r5s5+990r5s6+420r5s7+1728r6s6+1092r6s71,1=8060,ND5G3=136r2s3+52r2s4+2512r3s4+3415r3s5+52r3s6+34r4s4+263r4s6+2r5s5+6110r5s6+7435r5s7+8r6s6+8521r6s71,1=32947420,ISI´G3=65r5+43r6+127r7+2878r8+2r9+12110r10+6011r11+8413r13+17912r121=79.303,H′G3=25r5+13r6+27r7+92r8+29r9+r10+611r11+56r12+413r131=8.627,SG3=8r3+8r4+1728125r5+870427r6+5832343r7+70921512r8+10009r9+2294991000r10+1481761331r111=1005.794.


## 4. Conclusion

Monkeypox has re-emerged. There is no direct vaccine or medicine for this disease. Vaccines and drugs used for smallpox are used as treatment of this disease. Cost and time are important in drug and vaccine discovery. Molecular descriptors are used to reduce this process. In this study,M-polynomials and NM-polynomials of drugs approved for monkeypox are obtained. Using these polynomials, some molecular descriptors that depend on the degree and some molecular descriptors that depend on the sum of their neighborhood degrees of these drugs are obtained. The results of this study can be used for QSPR and QSAR studies to help predict the properties of drugs and vaccines to be discovered for the treatment of monkeypox. Figures 4–6 show plots of M-polynomials and NM-polynomials of tecovirimat, cidofovir, and brincidofovir, respectively.Figure 4
(a)M-polynomial of tecovirimat. (b) NM-polynomial of tecovirimat.
(a)(b)Figure 5
(a)M-polynomial of cidofovir. (b) NM-polynomial of cidofovir.
(a)(b)Figure 6
(a)M-polynomial of brincidofovir. (b) NM-polynomial of brincidofovir.
(a)(b)From Figure4, it can be seen that the degree-based topological indices of tecovirimat drug grow slower and more regularly than the topological indices based on the sum of the neighborhood degree.From Figure5, the NM-polynomial of the drug cidofovir appears to grow much faster than the degree-based polynomial.It is seen from the plots in Figure6 that the M-polynomial of the brincidofovir drug has smaller values compared to the NM-polynomials.



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*Source: 9971255-2022-11-17.xml*