Arabian Journal of Chemistry (2021) 14, 103240

King Saud University

Arabian Journal of Chemistry

www.ksu.edu.sa
www.sciencedirect.com

ORIGINAL ARTICLE

Quantitative structure-property relationships

(QSPR) of valency based topological indices with
Covid-19 drugs and application
Jian-Feng Zhong a, Abdul Rauf b,*, Muhammad Naeem b, Jafer Rahman c,
Adnan Aslam d

a
Department of Infectious Diseases, Huzhou Central Hospital, Zhejiang, Huzhou 313000, PR China
b
Department of Mathematics, Air University Multan Campus, Multan, Pakistan
c
Department of Mathematics & Statistics, Hazara University Mansehra, Pakistan
d
Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore, Pakistan (RCET), Pakistan

Received 13 April 2021; accepted 27 May 2021

Available online 8 June 2021

KEYWORDS Abstract The purpose of this analysis is to establish a quantitative structure–property relation-
Ev-degree; ship (QSPR) between eV and ve-degree based topological descriptors and measured physico-
Ve-degree; chemical parameters of phytochemicals screened against SARS-CoV-2 3CLpro . A computer-
Topological indices; based algorithm is developed to compute the eV and ve-degree based topological indices for
Quantitative Structure- the considered graphs. Our study revealed that the eV-degree based Zagreb index Mev and
Property Relationships ve-degree based first beta Zagreb index Mbve1 are two important topological indices that can
(QSPR); be useful in the prediction of molecular weight and the topological polar surface area of phy-
Camptothecin-Polymer Con- tochemicals. Applications to certain anticancer drug (Camptothecin-Polymer Conjugate IT-101)
jugate IT-101; are presented at the end.
Ó 2021 The Author(s). Published by Elsevier B.V. on behalf of King Saud University. This is an open
05C09;
access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
05C92;
92E10

1. Introduction in the modeling of the molecular structure of chemical com-

pounds and also to study their chemical and physical
Chemical graph theory is a branch of science which deals with properties. Due to its wide-ranging applications in various
the graphical representation of the chemical structure. In fields of life such as electrical networks, biological networks,
chemistry, the chemical compounds that have the same molec- chemistry, computer science and drug designs, it attains much
ular formula but different structure are called isomers. In attention of researchers (Hosamani et al., 2017; Li et al., 2021;
chemical graph theory, the vertices represent the atoms and Shao et al., 2018). Recently, a mixture field of information
the edges represent the bonds. Chemical graph theory is used science, chemistry and mathematics have been developed, the
so-called chem-informatics.
* Corresponding author.
Several viral diseases continue to emerge causing serious
public health issue. Among these viral diseases severe acute
E-mail addresses: attari_ab092@yahoo.com, abdul.rauf@aumc.edu.
pk (A. Rauf), jaferattari@hu.edu.pk (J. Rahman). respiratory coronavirus syndrome (SAR-CoV) were reported

https://doi.org/10.1016/j.arabjc.2021.103240
1878-5352 Ó 2021 The Author(s). Published by Elsevier B.V. on behalf of King Saud University.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
2 J.-F. Zhong et al.

in 2002 and 2003 and H1N1 influenza in 2009. A fatal respira- 2021 the Moderna COVID-19 vaccine (mRNA 1273) was
tory disease as a result of novel coronavirus strain was listed in WHO Emergency Use Listing (EUL). On May 7,
reported at the end of 2019 (Xu et al., 2020). Soon after that 2021 the Sinopharm COVID-19 vaccine was listed for EUL
researcher, Chinese health authorities and Centers for Disease and was produced by the China National Biotec Group. Glob-
Control and Prevention (CDC) had taken swift action against ally, 165,772,430 confirmed COVID-19 cases were registered to
the disease. WHO temporarily named this virus as a novel the WHO till May 22, 2021 including 3,437,545 deaths. Vari-
coronavirus (2019-nCoV) (Ji et al., 2020). ous clinicians and researchers are focusing on research and
The first complete 2019-nCoV genome sequence was pub- production of antivirals using various methods that combine
lished on 10 January 2020. On 12 February 2020 the WHO experimental and in silico (Kumar et al., 2016 Jul 1; Mittal
classified the 2019-nCoV pathogen permanently as SARS- et al., 2019; Muralidharan et al., 2020; Nutho et al., 2020;
CoV-2 and coronavirus disease 2019 as COVID-19. This out- Needle et al., 2015; Pant et al., 2020; Pushpakom et al.,
break was officially declares as pandemic on March 11, 2020 2019; ul Qamar et al., 2020). The SARS-CoV-2 replication
by WHO. Typically, after transcription of the genome, beta- cycle can be broken down into three steps: viral RNA replica-
coronaviruses generate 800 k Da polypeptide. Proteolytic tion, viral entry and viral assembly and evacuation from the
clamping for several proteins is achieved. Proteolytic treatment host cell, as shown in Fig. 1.
is mediated by papain-like protease (PLpro ) and 3- The genome sequence of SARS-CoV-2 was found to be
chymotrypsin-like protease (3CLpro). Potential inhibitors very similar to that of SARS-CoV in recent studies. SARS-
have been identified for SARS-CoV and MERS-CoV 3CLpro CoV-2; 3CLpro -screened phytochemicals as given in Figs. 2–7
based on the structure of activity tests and high throughput are recently published in (ul Qamar et al., 2020) and several
studies (Ghosh et al., 2005; Kumar et al., 2016; Pillaiyar researchers are working to find better and productive instru-
et al., 2016). This research is therefore carried out to gain ments and medicines to combat diseases.
structural insights of SARS-CoV-2; 3CLpro and to detect pow- Topological indices are often characterized by using vertex
erful natural compounds to battle COVID-19. and edge degree-based concepts and play an important role in
The first emergency use (EUA) vaccine for COVID-19 was theoretical chemistry. The first topological descriptor was put
released by Food and Drug Administration (FDA) On Decem- forward by Wiener (1947) and is related with the critical point,
ber 11, 2020. WHO listed Pfizer/BioNtech Comirnaty vaccine boiling points, and density of paraffin. The Randic index was
in Emergency Use Listing (EUL) on December 31, 2020. On proposed by Randic in 1975 (Randic, 1975) and generalized by
February 16, 2021 the SII/Covishield and AstraZeneca/ Bollobás and Erdös (1998). The first and second Zagreb index
AZD1222 vaccines, on March 12, 2021 the Janssen/Ad26. was introduced by Gutman and Trinajstic (1972) about forty
COV 2.S developed by Johnson & Johnson and on April 30, years ago. The atom-bond connectivity (ABC) index was initi-

Fig. 1 SARS-COV-19 replication cycle.

Quantitative structure-property relationships (QSPR) of valency based topological indices 3

Fig. 2

Fig. 3

Fig. 4

Fig. 5
4 J.-F. Zhong et al.

Fig. 6

Fig. 7

ated by Estrada et al. (1998) and geometric-arithmetic index et al. calculated the reverse indices for remdesivir (GS-5734)
was introduced by Vukičević and Furtula (2009). The ve- (Wei et al., 2021). Kirmani, Syed Ajaz K., Parvez Ali, and Fai-
degree and eV-degree definitions were introduced by Chellali zul Azam investigated several antiviral drugs and QSPR was
et al. (2017). The work of Chellali et al. was investigated by established between topological indices and various physical/-
Horoldagva (2019) and mathematical concepts were devel- chemical properties of antiviral drugs (Kirmani et al., 2021).
oped. The degree-based ideology transformed into ve-degree, In the QSPR study, the bioactivity of chemical compounds
eV-degree, degree-based M-polynomial and NM-polynomials can be predicted by using topological indices. Shirakol et al.
and degree-based entropy. Various researchers calculated the (2019) study the QSPR analysis of degree-distance and dis-
different topological indices for COVID drugs-related struc- tance based topological indices for their predicting power.
tures, for the help of the production of antivirals (Rauf Similarly, Luccic et al. (2001) study the QSPR for novel
et al., 2021; Al-Ahmadi et al., 2021; Saleh et al., 2021). Liu distance-related indices. Using modified Xu and atom-type-
et al. (2021) studied the structural properties by using bond based AI topological indices a QSPR was performed for the
additive and distance based topological descriptors of antiviral prediction of enthalpies of 134 acyclic alkanes by Safa and
medications for the treatment of COVID 19 such as hydroxy- Yekta (2017). H. Sunilkumar et al. studied the QSPR analysis
chloroquine, chloroquine, lopinavir, theaflavin, ritonavir, of degree-based topological indices to characterize the useful
remdesivir, nafamostat, umifenovir, bevacizumab, and camo- topological indices based on their predicting power
stat. Topological indices of chloroquine, theaflavin, remdesivir (Hosamani et al., 2017). For more details on QSPR study of
and hydroxychloroquine are computed in (Mondal et al., topological indices see (Mondal et al., 2021; Sahoo et al.,
2020). Nandini, G. Kirithiga, et al. have computed topological 2011). The main objective of this paper is to establish a relation
indices of pandemic trees, corona product of Christmas trees between topological descriptors and physical/chemical param-
and paths (Nandini et al., 2020). Mondal, Sourav, et al. calcu- eters of phytochemicals that are used for screening against
lated the multiplicative degree-based indices for some anti- SARS-CoV-2; 3CLpro in a quantitative structure–property. In
COVID-19 chemicals such as hydroxychloroquine, theaflavin this paper, we have considered five ve and eV-degree based
and remdesivir (GS-5734) (Mondal et al., 2020). Wei, Jianxin, ß ahin1 and Ediz, 2018).
topological indices see (S
Quantitative structure-property relationships (QSPR) of valency based topological indices 5

Let G be a connected graph with vertex set and edge set (Camptothecin-Polymer Conjugate IT-101) is presented in Sec-
denoted by VðGÞ and EðGÞ respectively. For any v 2 VðGÞ, tion 5. Section 6 is the numerical results and discussion and
let KðvÞ represent the degree of the vertex v 2 VðGÞ and is Section 7 is conclusion. Section 3 describes an illustrative
the number of edges linked to v. The open neighborhood of example to demonstrate the algorithm described in Section 2.
the vertex v is the set of all vertices that are adjacent to v.
The closed neighborhood of v, denoted by N½v is defined as 2. Algorithm
the union of v vertex with open neighborhood of v. The eV-
degree of any edge e ¼ uv 2 EðGÞ, denoted by Kev ðeÞ is the
total number of the vertices of closed neighborhoods of the
end vertices of an edge e. The ve-degree of any vertex
v 2 VðGÞ, denoted by Kve ðvÞ is the total number of edges which
1: Start
are adjacent to v and the first neighbor of v, i.e., the sum of
2: Input: M is adjacency matrix of a Graph.
degrees of all closed neighbourhood vertices of v. 3: Output: Calculation of Ve-Degree, Ev-degree, Ve-degrees of
Ev-degree based indices end vertices of each edge, and the Ev and Ve degree based
The indices based on eV-degree, such as the Zagreb index topological indices.
(Mev ) and the Randic index (Rev ) for any edge e ¼ uv 2 EðGÞ 4: Initialization: E No. of edges, V No. of vertex, con[E]
are defined as connection matrix, deg[V] degree of each vertex, VE VE-
X Degree of vertices, deg [VE] No. of edges incident to the closed
Mev ðGÞ ¼ Kev ðeÞ2 ; neighborhood vertices, EV EV-Degree of adjacent vertices, deg
e2E [EV] the sum of the degree of two adjacent vertices, ver[V]
X Vertex list, count 1, adj[count] adjacent element, Mq
Kev ðeÞ2 :
1
Rev ðGÞ ¼ [count] VE-Degree matrices elements, Mp[count] EV-Degree
e2E matrices elements.
5: loop a = 1 to V
Ve-degree based index 6: For each vertex from the array ver[V].
For any vertex v 2 VðGÞ, the first Zagreb alpha index (Mave
1 ) 7: loop b = 1 to E
based on ve-degree is defined as 8: count corresponding vertex from the matrix con[E].
X 9: b++
Mave
1 ðGÞ ¼ Kve ðvÞ2 : 10: end loop
v2V 11: deg[V] = count.
Ve-degree of end vertices of each edge 12: loop c = 1 to count
13: adj[count] = store corresponding vertex.
The first and second Zagreb beta index denoted by Mbve1 14: c++
and Mbve
2 respectively of each edge uv 2 EðGÞ based on ve- 15: end loop
degree of end vertices are defined as 16: loop d = 1 to deg[V]count
X 17: Multiply Matrix M and adj[count]
Mbve
1 ðGÞ ¼ ðKve ðuÞ þ Kve ðvÞÞ; 18: Store the result of the above as VE
uv2E 19: d++
X 20: end loop
Mbve
2 ðGÞ ¼ ðKve ðuÞ  Kve ðvÞÞ: 21: loop e = 1 to V
uv2E 22: loop f = 1 to V
23: Summation of deg[V][e] with deg[V][f] and multiply
The distribution of the sections has the following details, see by (i,j) th element of Adjacency Matrix.
the procedure detail in Fig. 8. Section 2 describe the schematic 24: Store the result of the above as EV
computer-based algorithm to calculate the degree vector, ve- 25: f++
degree vector, and eV-degree matrix and the respective topo- 26: end loop
logical indices for a given graph through Maple software. 27: e++
The Section 4 describes the quantitative-structure–property 28: end loop
analysis (QSPR) of phytochemicals screened against SARS- 29: end loop
CoV-23CLpro with the help of eV and ve degree-based topolog- 30: loop g = 1 to VE
31: For each vertex from the array VE
ical descriptors. Applications to certain anticancer drug
32: loop h = 1 to VE
33: count vertex from the matrix for VE
34: end loop
35: deg[VE]=count
36: loop k = 1 to count
37: Mq[count]= store vertices for VE.
38: k++
39: end loop
40: end loop
41: loop l = 1 to EV
42: For each adjacent vertex from the array EV
43: loop m = 1 to EV
44: count corresponding vertices from the array EV

(continued on next page)

Fig. 8 Working flowchart.
6 J.-F. Zhong et al.

45: m++
46: end loop
47: deg[EV]=count
48: loop n = 1 to count
49: Mp[count]=store EV
50: n++
51: end loop
52: end loop
53: loop o = 1 to count
54: Calculate the VE-Degree based first Zagreb Alpha index.
55: end loop
56: loop p = 1 to count
57: Calculate the EV-Degree based Zagreb Index and Randic
Index.
58: end loop
59: loop q = 1 to E Fig. 9 Molecular structure of starphene for n ¼ 2; m ¼ 2 and
60: Calculate the End Vertices Ve degree for each Edge, the
l ¼ 3.
first and second Zagreb beta index based on the ve-degree of end
vertices of each edge, the first Zagreb alpha index based on ve-
degree, the Zagreb and Randic index based on eV-degree. corresponding row and we get the degree sequence of star-
61: end loop phene graph S in the form of row matrix B.
62: end
B ¼ ½2; 2; 2; 3; 3; 2; 2; 2; 2; 3; 3; 2; 2; 2; 2; 3; 3; 2
Note that the degree sequence is according to the vertices
labeled in Fig. 9. For Ve-degree of vertices, we multiplied the
a, b, c, d, e, f, g, h, k, l, m, n, o, p and q all are variables
adjacency matrix A and B. The column matrix AB represent
whose data type is an integer. We are using these variables
the Ve-degree of vertices.
for iterating through the matrix using indices in a loop.
AB ¼ ½4; 4; 5; 8; 8; 5; 4; 4; 5; 8; 8; 5; 4; 4; 5; 8; 8; 5
3. Illustrative example on Algorithm Let Aij denote the entry in the i-th row and j-th column of adja-
cency matrix A and Bi denoted the i-th entry of row matrix B.
We calculated the degree vector, ve-degree vector, and eV- We multiply the entry Aij of adjacency matrix A with the entry
degree matrix and the respective topological indices for a given Bi þ Bj to obtain a entry Lij of matrix L. The nonzero entries in
graph through the Maple algorithm given above. For this we the upper/lower triangular form of this matrix L give the Ev-
consider the example of starphene graph (S) for n ¼ 2; m ¼ 2 degree list of the given graph.
and l ¼ 2, see Fig. 9. First, we write the adjacency matrix 2 3
0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4
(A) of the graph S in MAPLE. We get the adjacency matrix 6 7
60 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 07
A by newGraph software (procedure mentioned in (Hayat 6 7
60 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 07
6 7
and Khan, 2021)). As there are 18 vertices in the graph, the 6 7
60 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 6 07
order of the A will be 18–18 and from figure it is clear that 6 7
60 0 0 0 0 5 0 0 0 6 0 0 0 0 0 0 0 07
6 7
the graph S has 21 edges. 6 7
60 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 07
2 3 6 7
60 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 07
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 7
6 7 6 7
61 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 07 60 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 07
6 7 6 7
60 0 07
60
6 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 077 6 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 7
6 7 L¼6 7
60 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 07 60 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 07
6 7 6 7
60 0 07
60
6 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 077 6 0 0 0 0 0 0 0 0 0 0 5 0 0 0 6 7
6 7 6 7
60 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 07 60 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 07
6 7 6 7
60 0 07
60
6 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 077 6 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 7
6 7 6 7
60 07 60 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 07
6
0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0
7 6 7
60 0 07
60
6 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 077 6
6
0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 7
7
A¼6 7 60
60 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 07 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 077
6 7 6 7
60
6 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 077 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 55
6 7
60
6 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 077
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
60
6 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 077
6 7 We use separate formula commands against each index. The cal-
60
6 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 077 culated values of the topological indices for the given graph are
60
6 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 077
6 7
60
6 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 077 Mev ðSÞ ¼ 510
6 7
40 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 15 Mave
1 ðSÞ ¼ 630
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
Mbve
1 ðSÞ ¼ 252
Now, with the help of the adjacency matrix we can calculate Mbve
2 ðSÞ ¼ 792
the degree of vertices, Ve-degree and Ev-degree. The degree pffiffi pffiffiffiffi

of a vertex can be obtained by adding the entries of the Rev ðSÞ ¼ 3 5

5
þ 3 1010 þ 32
Quantitative structure-property relationships (QSPR) of valency based topological indices 7

4. Quality testing analysis results are highly significant at 1% level of significance. This
shows that there exists a strong positive linear bivariate rela-
In this section, we study the QSPR of phytochemicals that tionship in the group of variables. The same extent of corre-
are used for screening against SARS- CoV-23CLpro using lation is depicted in the Figs. (10) and (11). The table
eV and ve degree-based topological descriptors. The produc- suggests that molecular weight (MW) and topological polar
tivity of the topological indices listed above was checked surface (TPS) depend on other indices such as
bve
using the phytochemical data set contained in (Balaban, Mev ; Rev ; Mave
1 ; M1 and Mbve
2 . Keeping in mind the above
1982) and https://pubchem.ncbi.nlm.nih.gov/. The data con- relationships, we consider the following linear regression
tains the information of the following variables: binding model for prediction analyses. (see Fig. 12)
affinity, molecular weight, docking score and topological P ¼ a þ bTI
polar surface. IDs of phytochemicals are mentioned in Table 1
and their properties in Table 2. We could not found topolog- Where P consists of either MW or TPS and TI denotes a pre-
bve bve
ical polar surface and molecular weight of NPACT00105. dictor in the list (Mev ; Rev ; Mave
1 ; M1 ; M2 ).
Therefore, we do not include NPACT00105 for QSPR. Ve- Based on the data in Table 3 with 12 observations in each
degree and eV-degree based topological indices are presented variable, we have the following estimated regression equations
in Table 3. Table 4 shows the results of correlation coeffi- which can be used to predict molecular weight for a given
bve
cients between each pair of variables. We note that all the value of any index Mev ; Rev ; Mave1 ; M1 and Mbve
2 .

Table 1 Phytochemical Name and PubChem IDs.

PhytochemicalName PubChemIDs
0 0 0
5; 7; 3 ; 4  Tetrahydroxy  2  1610052
ð3; 3  dimethylallylÞisoflavone

Myricitrin 5281673
Methylrosmarinate 6479915
3; 5; 7; 30 ; 40 ; 50  hexahydroxyflavanone NPACT00105
3  O  beta  D  glucopyranoside

ð2SÞ  Eriodictyol7  O  ð600  O  galloylÞ 10930068

beta  D  glucopyranoside

CalceolariosideB 5273567
Myricetin3  O  beta  D  glucopyranoside 5318606
Licoleafol 1111196
Amaranthin 6123095
Nelfinavir 64143
Prulifloxacin 65947
Colistin 5311054

Table 2 Properties of phytochemical.

PubChem Docking Binding Molecular Tolpological Polar
IDs Score Affinity Weight Surface

1610052 16.35 29.57 354.40 107

5281673 15.64 22.13 464.40 207
6479915 15.44 20.62 374.30 134
NPACT00105 14.42 19.10 0.00 00
10930068 14.41 19.47 602.50 253
5273567 14.36 19.87 478.40 186
5318606 13.70 18.42 480.40 227
1111196 13.63 19.64 372.40 127
6123095 12.67 18.14 726.60 346
64143 12.20 17.31 567.80 127
65947 11.32 15.40 461.50 125
5311054 13.73 18.57 1155.40 491
8 J.-F. Zhong et al.

Table 3 Ve-degree and Ev-degree based Topological Indices.

PubChem IDs Mev Rev Mave
1 Mbve Mve
2
1

11610052 1306 29.5225 1532 590 1954

5281673 1664 50.0469 2005 766 2699
6479915 1270 20.7257 1505 578 1855
NPACT00105 1532 22.9972 1857 700 2359
10930068 2190 32.5014 2110 1226 4862
5273567 1802 27.2264 2152 816 2784
5318606 1736 24.5579 1902 726 2404
11111496 1450 18.4823 1748 591 2197
6123095 2909 41.731 3483 1284 4822
64143 3000 37.4362 3602 1139 4801
605947 1804 24.0714 2133 804 2943
5311054 6116 76.8146 7152 2638 9107

Table 4 Correlation matrix.

MW TPS Mev Rev Mave
1 Mbve Mbve
1 2

MW 1.000 0.934 0.974 0.873 0.961 0.987 0.970

TPS 1.000 0.846 0.816 0.826 0.886 0.847
Mev 1.000 0.880 0.996 0.986 0.975
Rev 1.000 0.880 0.883 0.861
Mave
1 1.000 0.969 0.955
Mbve
1
1.000 0.989
Mbve
2
1.000

MW ¼ a þ bX TPS ¼ 53:422 þ 0:070Mev

MW ¼ 189:809 þ 0:159M ev
TPS ¼ 21:712 þ 5:572Rev
MW ¼ 148:351 þ 11:687Rev TPS ¼ 58:694 þ 0:058Mave
1
MW ¼ 198:166 þ 0:133Mave
1 TPS ¼ 39:863 þ 0:172Mbve
1
MW ¼ 171:654 þ 0:377Mbve
1 TPS ¼ 47:565 þ 0:046Mbve
2
MW ¼ 178:179 þ 0:103Mbve
2

Table 6 gives the results of various statistics for bivariate

bve
Table 5 gives the results for correlation coefficient (r), coef- regression equations of TPS on either of Mev ; Rev ; Mave 1 ; M1
bve
ficient of determination (R2 ), F-statistic (F) and standard error and M2 . We observe that the estimated regression of TPS
of the estimates (s) for simple linear regression models between on Mbve1 possesses highest values of coefficient of determination
MW and various predictors. The results exhibit that the esti- and F-statistic (R2 = 0.7850, F = 36.529) and least value of
mated model of MW on Mbve 1 consists of highest coefficient standard error of regression (s = 54.4280). The results reveal
of determination and F-value (R2 = 0.9742, F = 375.812) that the regression model (TPS on Mbve 1 ) is highly significant.
and least standard error (s = 37.0609). This means that the TPS is highly determined by only Mbve and the estimated line
1
model is highly significant. The only index Mbve has a high fits the observations closer than other lines. Hence, Mbve
1 seems
1
influence on MW and the observed values fall closer to the fit- to the best predictor of topological polar surface as well. The
ted line. Thus, Mbve
1 is the best predictor of molecular weight. hierarchical order of the indices according to their perfor-
Similarly, Mev predicts molecular weight better than others mance is shown in Table 7.
after Mbve
1 . A priority list of indices according to their perfor- Based on correlation and regression analyses, we may con-
mance is given in Table 7. clude that both molecular weight and the topological polar
Following are the estimated models for predicting topolog- surface can be predicted well by the indices in the order men-
ical polar surface for a given value of any index tioned in Table 7. However, these indices cannot determine the
bve bve
Mev ; Rev ; Mave
1 ; M1 and M2 . The estimation results are based
other variables well such as docking scores (DS) and binding
on the data in Table 3 with 12 observations in each variable. affinity (BA). Table 8 shows the results of the bivariate corre-
Quantitative structure-property relationships (QSPR) of valency based topological indices 9

Fig. 10 Graphical representation of linear regression model between topological indices and molecular weight.

lation coefficient which are all insignificant. This means that 5. Camptothecin-Polymer Conjugate IT-101
DS and BA are not linearly correlated with all the indices
bve
Mev ; Rev , Mave
1 ; M1 and Mbve
2 . Thus, linear regression cannot Camptothecin (CA) is an alkaloid with a unique anticancer
model their relationship. function, operates with a vital and unique mechanism of
10 J.-F. Zhong et al.

Fig. 11 Graphical representation of linear regression model between topological indices and topological polar surface.

the action to target the topoisomerase (I) nuclear enzyme. limitations due to severe toxicity. Camptothecin shows potent
The development of a large range of tumors is inhibited by in vitro anti-glioma activity, making it another common poly-
CA. IT-101 is a conjugate of camptothecin with a mer delivery candidate. In the 9L gliosarcoma rat model,
cyclodextrin-based polymer (Schluep et al., 2006). Topoiso- sodium salt of camptothecin loaded into 50 percent polymers
merase I is affected by a drug known as Camptothecin, was checked and resulted in a substantial survival increase
derived from the Chinese tree, that allows the DNA cleavage, (Thomas et al., 2004). The median survival with camp-
but it inhibits the subsequent ligation and breaks the DNA tothecin polymers was nineteen days in the model of control
chain. The systemic application of the camptothecin has some rats and up to 120 days. Compared with controls, when we
Quantitative structure-property relationships (QSPR) of valency based topological indices 11

Fig. 12 Molecular Structure of IT-101. The components of the parentpolymer are polyethylene glycol and b-cyclodextrin. Camptothecin
is attached to the polymer via a single glycine amino acid linker, where n and m represent the number of repeating units of ethylene glycol
and cyclodextrin-based polymer-camptothecin in the polymer-camptothecin conjugate respectively.

Table 5 Performance measures.

X r R2 F s
Mev 0.9737 0.9481 182.3956 52.4815
Rev 0.8729 0.762 31.996 112.3313
Mave
1 0.9606 0.9228 119.339 64.0086
Mbve
1
0.987 0.9742 375.812 37.0609
Mbve
2
0.9705 0.9419 161.769 55.5432

Table 6 Performance measures.

X r R2 F s
Mev 0.8458 0.7154 25.140 62.6308
Rev 0.8160 0.6659 19.925 67.8686
Mave
1 0.8259 0.6821 21.457 66.1960
Mbve
1
0.8860 0.7850 36.529 54.4280
Mbve
2
0.8467 0.7169 25.324 62.4670

used direct intratumoral injection of our drug camptothecin

Table 7 Correlation coefficient and F Values. without polymer then it showed no betterment in the survival
Index Priority-wise Position (Liu et al., 2000). Studies integrating camptothecin into 50%
polymers for the treatment of 9L gliosarcoma in the rats. It
Mbve
1
1
gives protection and efficacy with the survival of 69 days in
ev
M 2 animals that were treated with the drug camptothecin. Under
Mbve
2
3 similar conditions, polymer administration of intracranial
Mave
1 4 camptothecin showed a substantial betterment in survival as
Rev 5 compared with the alone camptothecin (Schultz, 1973). In
12 J.-F. Zhong et al.

Table 8 Insignificant correlations.

DS BA Mev Rev Mave
1 Mbve Mbve
1 2

DS 1.000 0.840 0.311 0.015 0.322 0.257 0.325

BA 1.000 0.306 0.039 0.304 0.285 0.329

Table 9 The edges eV-degree of Camptothecin-Polymer Conjugate IT-101.

ðKðuÞ; KðvÞÞ Kev ðeÞ Frequency
Eð1; 2Þ 3 19m
Eð1; 3Þ 4 25m þ 1
Eð1; 4Þ 5 4mn þ 85m þ 6
Eð2; 3Þ 5 8m
Eð2; 4Þ 6 mn þ 60m
Eð3; 3Þ 6 35m
Eð3; 4Þ 7 20m þ 2
Eð4; 4Þ 8 mn þ 43m

our work, we compute five ve-degree and eV-degree based Table 10 The vertex ve-degree of Camptothecin-Polymer
topological indices (Chen et al., 2021) of Camptothecin- Conjugate IT-101
Polymer Conjugate IT-101. The vertex set of
Camptothecin-Polymer Conjugate IT-101 can be partitioned KðuÞ Kve ðuÞ Frequency
in four set based on the degree of vertices. There are 1 2 19m
4mn þ 129m þ 7 vertices of degree 1; mn þ 33m vertices of 1 3 25m þ 1
degree 2; 43m þ 1 vertices of degree 3 and 2mn þ 62m þ 2 1 4 4mn þ 85m þ 6
vertices of degree 4. Similarly, the Edge partition Eði; jÞ of 2 5 19m
2 6 2m
IT-101 based on the degree of end vertices i and j of an edge
2 7 4m
e ¼ ij is as follows; Eð1; 2Þ with 19m edges, Eð1; 3Þ with 2 8 mn þ 8m
25m þ 1 edges, Eð1; 4Þ with 4mn þ 85m þ 6 edges, Eð2; 3Þ with 3 7 18m
8m edges, Eð2; 4Þ with mn þ 60m edges, Eð3; 3Þ with 35m 3 8 12m
edges, Eð3; 4Þ with 20m þ 2 edges and Eð4; 4Þ with mn þ 43m 3 9 5m þ 1
edges. 3 10 8m
4 6 2
4 7 4m
Theorem 1. Let H be a molecular graph of Camptothecin-
4 8 2mn þ 15m
Polymer Conjugate IT-101 structure, then eV-degrees based 4 9 9m
Zagreb index and eV-degree based Randic index are given by, 4 10 2m
(a) Mev ðHÞ ¼ 200mn þ 10048m þ 264. 4 11 30m
4 12 2m
(b) Rev ðHÞ ¼ ðp4ffiffi5 þ p1ffiffi6 þ p1ffiffi8Þmnþ ðp19ffiffi3 þ p25ffiffi4 þ p85ffiffi5 þ p8ffiffi5 þ
60ffiffi
p
6
þ p35ffiffi þ p20ffiffi þ p43ffiffiÞm.
6 7 8
ðp1ffiffi þ p6ffiffi þ p2ffiffiÞ.
4 5 7
(b) The Randic index
X
Kev ðeÞ2 ;
1
Proof. By using the definition, we have calculated the eV- R ðHÞ ¼
ev

e2EðHÞ
degree of the each edge partition as shown in Table 1.
ð3Þ2 jEð1;2Þ j þ ð4Þ2 jEð1;3Þ j þ ð5Þ2 jEð1;4Þ j þ ð5Þ2 jEð2;3Þ j þ ð6Þ2 jEð2;4Þ j
1 1 1 1 1
Rev ðHÞ ¼
12 12 12
From Table 9, we can calculate the eV-degree based indices þ ð6Þ jEð3;3Þ j þ ð7Þ jEð3;4Þ j þ ð8Þ jEð4;4Þ j
¼ ðp4ffiffi þ p1ffiffi þ p1ffiffiÞmn þ ðp19ffiffi þ p25ffiffi þ p85ffiffi þ p8ffiffi þ p60ffiffi þ p35ffiffi þ p20ffiffi þ p43ffiffiÞm
such as: (see Table 10) 5 6 8 3 4 5 5 6 6 7 8
þ ðp1ffiffi4 þ p6ffiffi5 þ p2ffiffi7Þ:
(a) The Zagreb index
X 2
Mev ðHÞ ¼ Kev ðeÞ ;
e2EðHÞ
2 2 2 2 2
Mev ðHÞ ¼ ð3Þ jEð1;2Þ j þ ð4Þ jEð1;3Þ j þ ð5Þ jEð1;4Þ j þ ð5Þ jEð2;3Þ j þ ð6Þ jEð2;4Þ j Theorem 2. Let H be a molecular graph of Camptothecin-
2 2 2
þ ð6Þ jEð3;3Þ j þ ð7Þ jEð3;4Þ j þ ð8Þ jEð4;4Þ j Polymer Conjugate IT-101 structure, then vertex ve-degree
¼ 200mn þ 10048m þ 264: based first Zagreb a-index is
Quantitative structure-property relationships (QSPR) of valency based topological indices 13
X
Mave
1 ðHÞ ¼ Kve ðvÞ2
Table 11 Ve-degree of end vertices of each edge of Camp- v2VðHÞ

tothecin-Polymer Conjugate IT-101 Mave 2 2 2 2 2 2

1 ðHÞ ¼ ð2Þ ð19mÞ þ ð3Þ ð25m þ 1Þ þ ð4Þ ð4mn þ 85m þ 6Þ þ ð5Þ ð19mÞ þ ð6Þ ð2mÞ þ ð7Þ ð4mÞ

þ þð8Þ2 ðmn þ 8mÞ þ ð7Þ2 ð18mÞ þ ð8Þ2 ð12mÞ þ ð9Þ2 ð5m þ 1Þ þ ð10Þ2 ð8mÞ
Edge ðKve ðuÞ; Kve ðvÞÞ Frequency þ ð6Þ2 ð2Þ þ ð7Þ2 ð4mÞ þ ð8Þ2 ð2mn þ 15mÞ þ ð9Þ2 ð9mÞ þ ð10Þ2 ð2mÞ
þ ð11Þ2 ð30mÞ þ ð12Þ2 ð2mÞ
E1 (2, 5) 19m
¼ 256mn þ 11774m þ 258:
E2 (3, 7) 18m
E3 (3, 8) 6m
E4 (3, 9) mþ1
E5 (4, 6) 6
E6 (4, 7) 10m Theorem 3. Let H be a molecular graph of Camptothecin-
E7 (4, 8) 4mn þ 30m Polymer Conjugate IT-101 structure, then ve-degree based
E8 (4, 9) 11m indices of end vertices of each edges are given by,
E9 (4, 10) 4m
E10 (4, 11) 30m
(a) Mbve
1 ðHÞ ¼ 80mn þ 4441m þ 105.
E11 (6, 8) 4m
(b) Mbve
2 ðHÞ ¼ 256mn þ 16160m þ 306.
E12 (7, 7) 4m
E13 (5, 8) 5m
E14 (5, 11) 14m Proof. According to definition of ve-degree of end vertices of
E15 (7, 7) 4m each edge, we divides the edges into 46 partitions i.e.,
E16 (7, 12) 2m E1 ; E2 E3 ; . . . ; E46 respectively, as shown in Table 3.
E17 (8, 8) mn þ 7m
E18 (8, 9) 14m
E19 (8, 11) 14m By using the table we can compute indices based on ve-
E20 (7, 7) 6m degree of end vertices of each edge as (a) The first Zagreb
E21 (7, 8) 2m b-index
E22 (7, 9) 6m X
E23 (7, 10) 8m Mbve
1 ðHÞ ¼ ðKve ðuÞ þ Kve ðvÞÞ
uv2EðHÞ
E24 (8, 8) 3m Mbve ð7ÞjE1 j þ ð10ÞjE2 j þ ð11ÞjE3 j þ ð12ÞjE4 j þ ð10ÞjE5 j þ ð11ÞjE6 j þ ð12ÞjE7 j
1 ðHÞ ¼
E25 (8, 9) 4m þ ð13ÞjE8 j þ ð14ÞjE9 j þ ð15ÞjE10 j þ ð14ÞjE11 j þ ð14ÞjE12 j þ ð13ÞjE13 j þ ð16ÞjE14 j
E26 (8, 10) 2m þ ð14ÞjE15 j þ ð19ÞjE16 j þ ð16ÞjE17 j þ ð17ÞjE18 j þ ð19ÞjE19 j þ ð14ÞjE20 j þ ð15ÞjE21 j
E27 (9, 10) 2m þ ð16ÞjE22 j þ ð17ÞjE23 j þ ð16ÞjE24 j þ ð17ÞjE25 j þ ð18ÞjE26 j þ ð19ÞjE27 j þ ð20ÞjE28 j
E28 (10, 10) 2m þ ð15ÞjE29 j þ ð19ÞjE30 j þ ð16ÞjE31 j þ ð17ÞjE32 j þ ð18ÞjE33 j þ ð19ÞjE34 j þ ð15ÞjE35 j
E29 (7, 8) 2m þ ð20ÞjE36 j þ ð17ÞjE37 j þ ð22ÞjE38 j þ ð17ÞjE39 j þ ð16ÞjE40 j þ ð17ÞjE41 j þ ð19ÞjE42 j
þ ð22ÞjE43 j þ ð22ÞjE44 j þ ð20ÞjE45 j þ ð18ÞjE46 j
E30 (7, 12) 2m
¼ ð7Þð19mÞ þ ð10Þð18mÞ þ ð11Þð6mÞ þ ð12Þðm þ 1Þ þ ð10Þð6Þ þ ð11Þð10mÞ
E31 (8, 8) 2m þ ð12Þð4mn þ 30mÞ þ ð13Þð11mÞ þ ð14Þð4mÞ þ ð15Þð30mÞ þ ð14Þð4mÞ þ ð14Þð4mÞ
E32 (8, 9) m þ ð13Þð5mÞ þ ð16Þð14mÞ þ ð14Þð4mÞ þ ð19Þð2mÞ þ ð16Þðmn þ 7mÞ þ ð17Þð14mÞ
E33 (10, 8) 4m þ ð19Þð14mÞ þ ð14Þð6mÞ þ ð15Þð2mÞ þ ð16Þð6mÞ þ ð17Þð8mÞ þ ð16Þð3mÞ þ ð17Þð4mÞ
E34 (8, 11) 3m þ ð18Þð2mÞ þ ð19Þð2mÞ þ ð20Þð2mÞ þ ð15Þð2mÞ þ ð19Þð2mÞ þ ð16Þð2mÞ þ ð17ÞðmÞ
E35 (9, 6) mþ1 þ ð18Þð4mÞ þ ð19Þð3mÞ þ ð15Þðm þ 1Þ þ ð20ÞðmÞ þ ð17Þð2mÞ þ ð22Þð2mÞ þ ð17Þð2mÞ
E36 (9, 11) m þ ð16ÞðmnÞ þ ð17Þð2mÞ þ ð19Þð9mÞ þ ð22Þð2mÞ þ ð22Þð21mÞ þ ð20Þð7mÞ þ ð18Þð1Þ
¼ 80mn þ 4441m þ 105:
E37 (10, 7) 2m
E38 (10, 12) 2m
(b) The second Zagreb b-index
E39 (7, 10) 2m X
E40 (8, 8) mn Mbve
2 ðHÞ ¼ ðKve ðuÞ  Kve ðvÞÞ
E41
uv2EðHÞ
(8, 9) 2m
Mbve
2 ðHÞ ¼ ð10ÞjE1 j þ ð21ÞjE2 j þ ð24ÞjE3 j þ ð27ÞjE4 j þ ð24ÞjE5 j þ ð28ÞjE6 j þ ð32ÞjE7 j
E42 (8, 11) 9m þ ð36ÞjE8 j þ ð40ÞjE9 j þ ð44ÞjE10 j þ ð48ÞjE11 j þ ð49ÞjE12 j þ ð40ÞjE13 j þ ð55ÞjE14 j
E43 (10, 12) 2m þ ð49ÞjE15 j þ ð84ÞjE16 j þ ð64ÞjE17 j þ ð72ÞjE18 j þ ð88ÞjE19 j þ ð49ÞjE20 j þ ð56ÞjE21 j
E44 (11, 11) 21m þ ð63ÞjE22 j þ ð70ÞjE23 j þ ð64ÞjE24 j þ ð72ÞjE25 j þ ð80ÞjE26 j þ ð90ÞjE27 j þ ð100ÞjE28 j
E45 (9, 11) 7m þ ð56ÞjE29 j þ ð84ÞjE30 j þ ð64ÞjE31 j þ ð72ÞjE32 j þ ð80ÞjE33 j þ ð88ÞjE34 j þ ð54ÞjE35 j
E46 (9, 9) 1 þ ð99ÞjE36 j þ ð70ÞjE37 j þ ð120ÞjE38 j þ ð70ÞjE39 j þ ð64ÞjE40 j þ ð72ÞjE41 j þ ð88ÞjE42 j
þ ð120ÞjE43 j þ ð121ÞjE44 j þ ð99ÞjE45 j þ ð81ÞjE46 j
¼ ð10Þð19mÞ þ ð21Þð18mÞ þ ð24Þð6mÞ þ ð27Þðm þ 1Þ þ ð24Þð6Þ þ ð28Þð10mÞ
þ ð32Þð4mn þ 30mÞ þ ð36Þð11mÞ þ ð40Þð4mÞ þ ð44Þð30mÞ þ ð48Þð4mÞ þ ð49Þð4mÞ
þ ð40Þð5mÞ þ ð55Þð14mÞ þ ð49Þð4mÞ þ ð84Þð2mÞ þ ð64Þðmn þ 7mÞ þ ð72Þð14mÞ
Mave
1 ðHÞ ¼ 256mn þ 11774m þ 258: þ ð88Þð7mÞ þ ð49Þð6mÞ þ ð56Þð2mÞ þ ð63Þð6mÞ þ ð70Þð8mÞ þ ð64Þð3mÞ þ ð72Þð4mÞ
þ ð80Þð2mÞ þ ð90Þð2mÞ þ ð100Þð2mÞ þ ð56Þð2mÞ þ ð84Þð2mÞ þ ð64Þð2mÞ þ ð72ÞðmÞ
Proof. By using the definition, the ve-degrees of the vertices þ ð80Þð4mÞ þ ð88Þð3mÞ þ ð54Þðm þ 1Þ þ ð99ÞðmÞ þ ð70Þð2mÞ þ ð120Þð2mÞ
þ ð70Þð2mÞ þ ð64ÞðmnÞ þ ð72Þð2mÞ þ ð88Þð9mÞ þ ð120Þð2mÞ þ ð121Þð21mÞ
are computed. This computation is presented Table 2.
þ ð99Þð7mÞ þ ð81Þð1Þ
By using the above table, we have a first ve-degree based ¼ 256mn þ 16160m þ 306:

Zagreb a-index.
14 J.-F. Zhong et al.

Table 12 Numerical results of indices for Camptothecin-Polymer Conjugate IT-101.

½m; n Mev ðHÞ Rev ðHÞ Mave
1 ðHÞ Mbve
1 ðHÞ
\color{black} {M}_{2}^{\beta ve}(H)

½1; 1 10512 133.0960589 12288 4626 16722

½2; 2 21160 267.3542194 24830 9307 33650
½3; 3 32208 406.7136921 37884 14148 51090
½4; 4 43656 551.1744769 51450 19149 69042
½5; 5 55504 700.7365737 65528 24310 87506
½6; 6 67752 855.3999827 80118 29631 106482
½7; 7 80400 1015.164704 95220 35112 125970
½8; 8 93448 1180.030738 110834 40753 145970
½9; 9 106896 1349.998082 126960 46554 166482
½10; 10 120744 1525.066740 143598 52515 187506

6. Numerical results

In this section, we will discuss the numerical results related to

the eV-degree and ve-degree based topological descriptors for
the Camptothecin-Polymer Conjugate IT-101 molecular struc-
tures. We have used different values of m and n and computed
numerical values for the eV-degree and ve-degree based indices
bve bve
such as Mev ðHÞ; Rev ðHÞ; Mave
1 ðHÞ; M1 ðHÞ and M2 ðHÞ for the
Camptothecin-Polymer Conjugate IT-101 structures. (see
Table 11).
We can see in Table 12 and Figs. 13–15 that value topolog-
ical descriptors increases when we increase the value of m and
n. The Zagreb types indices were found to occur for the com-
putation of the total p-electron energy of molecules; thus, for
higher values of m and n, the total p-electron energy is increas-
ing. The Randić index is used in the study of the chemical sim-
ilarity of molecular compounds and in computing the Kovats
Fig. 14 The first Zagreb a-index.
constants and boiling point of molecules. The results shows

Fig. 13 The eV-degrees based indices, (a) The Zagreb index, (b) The Randic index.
Quantitative structure-property relationships (QSPR) of valency based topological indices 15

Fig. 15 (a) The first Zagreb b-index, (b) The second b-Zagreb index.

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