Quantitative structure-property relationships (QSPR) of valency based topological indices with Covid-19 drugs and application

1 Introduction

Chemical graph theory is a branch of science which deals with the graphical representation of the chemical structure. In chemistry, the chemical compounds that have the same molecular formula but different structure are called isomers. In chemical graph theory, the vertices represent the atoms and the edges represent the bonds. Chemical graph theory is used in the modeling of the molecular structure of chemical compounds and also to study their chemical and physical properties. Due to its wide-ranging applications in various fields of life such as electrical networks, biological networks, chemistry, computer science and drug designs, it attains much attention of researchers (Hosamani et al., 2017; Li et al., 2021; Shao et al., 2018). Recently, a mixture field of information science, chemistry and mathematics have been developed, the so-called chem-informatics.

Several viral diseases continue to emerge causing serious public health issue. Among these viral diseases severe acute respiratory coronavirus syndrome (SAR-CoV) were reported in 2002 and 2003 and H1N1 influenza in 2009. A fatal respiratory disease as a result of novel coronavirus strain was reported at the end of 2019 (Xu et al., 2020). Soon after that researcher, Chinese health authorities and Centers for Disease Control and Prevention (CDC) had taken swift action against the disease. WHO temporarily named this virus as a novel coronavirus (2019-nCoV) (Ji et al., 2020).

The first complete 2019-nCoV genome sequence was published on 10 January 2020. On 12 February 2020 the WHO classified the 2019-nCoV pathogen permanently as SARS-CoV-2 and coronavirus disease 2019 as COVID-19. This outbreak was officially declares as pandemic on March 11, 2020 by WHO. Typically, after transcription of the genome, beta-coronaviruses generate 800 k Da polypeptide. Proteolytic clamping for several proteins is achieved. Proteolytic treatment is mediated by papain-like protease ( ${\mathit{PL}}^{\mathit{pro}}$ ) and 3-chymotrypsin-like protease (3CLpro). Potential inhibitors have been identified for SARS-CoV and MERS-CoV $3{\mathit{CL}}^{\mathit{pro}}$ based on the structure of activity tests and high throughput studies (Ghosh et al., 2005; Kumar et al., 2016; Pillaiyar et al., 2016). This research is therefore carried out to gain structural insights of SARS-CoV- $2,3{\mathit{CL}}^{\mathit{pro}}$ and to detect powerful natural compounds to battle COVID-19.

The first emergency use (EUA) vaccine for COVID-19 was released by Food and Drug Administration (FDA) On December 11, 2020. WHO listed Pfizer/BioNtech Comirnaty vaccine in Emergency Use Listing (EUL) on December 31, 2020. On February 16, 2021 the SII/Covishield and AstraZeneca/AZD1222 vaccines, on March 12, 2021 the Janssen/Ad26.COV 2.S developed by Johnson & Johnson and on April 30, 2021 the Moderna COVID-19 vaccine (mRNA 1273) was listed in WHO Emergency Use Listing (EUL). On May 7, 2021 the Sinopharm COVID-19 vaccine was listed for EUL and was produced by the China National Biotec Group. Globally, 165,772,430 confirmed COVID-19 cases were registered to the WHO till May 22, 2021 including 3,437,545 deaths. Various clinicians and researchers are focusing on research and production of antivirals using various methods that combine experimental and in silico (Kumar et al., 2016 Jul 1; Mittal et al., 2019; Muralidharan et al., 2020; Nutho et al., 2020; Needle et al., 2015; Pant et al., 2020; Pushpakom et al., 2019; ul Qamar et al., 2020). The SARS-CoV-2 replication cycle can be broken down into three steps: viral RNA replication, viral entry and viral assembly and evacuation from the host cell, as shown in Fig. 1.

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Fig. 1

SARS-COV-19 replication cycle.

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The genome sequence of SARS-CoV-2 was found to be very similar to that of SARS-CoV in recent studies. SARS-CoV- $2,3{\mathit{CL}}^{\mathit{pro}}$ -screened phytochemicals as given in Figs. 2–7 are recently published in (ul Qamar et al., 2020) and several researchers are working to find better and productive instruments and medicines to combat diseases.

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Fig. 2

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Fig. 3

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Fig. 7

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Topological indices are often characterized by using vertex and edge degree-based concepts and play an important role in theoretical chemistry. The first topological descriptor was put forward by Wiener (1947) and is related with the critical point, boiling points, and density of paraffin. The Randic index was proposed by Randic in 1975 (Randic, 1975) and generalized by Bollobás and Erdös (1998). The first and second Zagreb index was introduced by Gutman and Trinajstic (1972) about forty years ago. The atom-bond connectivity (ABC) index was initiated by Estrada et al. (1998) and geometric-arithmetic index was introduced by Vukičević and Furtula (2009). The ve-degree and eV-degree definitions were introduced by Chellali et al. (2017). The work of Chellali et al. was investigated by Horoldagva (2019) and mathematical concepts were developed. The degree-based ideology transformed into ve-degree, eV-degree, degree-based M-polynomial and NM-polynomials and degree-based entropy. Various researchers calculated the different topological indices for COVID drugs-related structures, for the help of the production of antivirals (Rauf et al., 2021; Al-Ahmadi et al., 2021; Saleh et al., 2021). Liu et al. (2021) studied the structural properties by using bond additive and distance based topological descriptors of antiviral medications for the treatment of COVID 19 such as hydroxychloroquine, chloroquine, lopinavir, theaflavin, ritonavir, remdesivir, nafamostat, umifenovir, bevacizumab, and camostat. Topological indices of chloroquine, theaflavin, remdesivir and hydroxychloroquine are computed in (Mondal et al., 2020). Nandini, G. Kirithiga, et al. have computed topological indices of pandemic trees, corona product of Christmas trees and paths (Nandini et al., 2020). Mondal, Sourav, et al. calculated the multiplicative degree-based indices for some anti-COVID-19 chemicals such as hydroxychloroquine, theaflavin and remdesivir (GS-5734) (Mondal et al., 2020). Wei, Jianxin, et al. calculated the reverse indices for remdesivir (GS-5734) (Wei et al., 2021). Kirmani, Syed Ajaz K., Parvez Ali, and Faizul Azam investigated several antiviral drugs and QSPR was established between topological indices and various physical/chemical properties of antiviral drugs (Kirmani et al., 2021).

In the QSPR study, the bioactivity of chemical compounds can be predicted by using topological indices. Shirakol et al. (2019) study the QSPR analysis of degree-distance and distance based topological indices for their predicting power. Similarly, Luccic et al. (2001) study the QSPR for novel distance-related indices. Using modified Xu and atom-type-based AI topological indices a QSPR was performed for the prediction of enthalpies of 134 acyclic alkanes by Safa and Yekta (2017). H. Sunilkumar et al. studied the QSPR analysis of degree-based topological indices to characterize the useful topological indices based on their predicting power (Hosamani et al., 2017). For more details on QSPR study of topological indices see (Mondal et al., 2021; Sahoo et al., 2011). The main objective of this paper is to establish a relation between topological descriptors and physical/chemical parameters of phytochemicals that are used for screening against SARS-CoV- $2,3{\mathit{CL}}^{\mathit{pro}}$ in a quantitative structure–property. In this paper, we have considered five ve and eV-degree based topological indices see (Şahin1 and Ediz, 2018).

Let G be a connected graph with vertex set and edge set denoted by $V(G)$ and $E(G)$ respectively. For any $v\in V(G)$ , let $\mathrm{\Lambda }(v)$ represent the degree of the vertex $v\in V(G)$ and is the number of edges linked to v. The open neighborhood of the vertex v is the set of all vertices that are adjacent to v. The closed neighborhood of v, denoted by $N[v]$ is defined as the union of v vertex with open neighborhood of v. The eV-degree of any edge $e=\mathit{uv}\in E(G)$ , denoted by ${\mathrm{\Lambda }}_{\mathit{ev}}(e)$ is the total number of the vertices of closed neighborhoods of the end vertices of an edge e. The ve-degree of any vertex $v\in V(G)$ , denoted by ${\mathrm{\Lambda }}_{\mathit{ve}}(v)$ is the total number of edges which are adjacent to v and the first neighbor of v, i.e., the sum of degrees of all closed neighbourhood vertices of v.

Ev-degree based indices

The indices based on eV-degree, such as the Zagreb index ( $M^{\mathit{ev}}$ ) and the Randic index ( $R^{\mathit{ev}}$ ) for any edge $e=\mathit{uv}\in E(G)$ are defined as $$M^{\mathit{ev}}(G)={\displaystyle \sum _{e\in E}}{\mathrm{\Lambda }}_{\mathit{ev}}{(e)}^2,$$ $$R^{\mathit{ev}}(G)={\displaystyle \sum _{e\in E}}{\mathrm{\Lambda }}_{\mathit{ev}}{(e)}^{-\frac{1}{2}}\text{.}$$

Ve-degree based index

For any vertex $v\in V(G)$ , the first Zagreb alpha index ( $M_1^{\alpha \mathit{ve}}$ ) based on ve-degree is defined as $$M_1^{\alpha \mathit{ve}}(G)={\displaystyle \sum _{v\in V}}{\mathrm{\Lambda }}_{\mathit{ve}}{(v)}^2\text{.}$$

Ve-degree of end vertices of each edge

The first and second Zagreb beta index denoted by $M_1^{\beta \mathit{ve}}$ and $M_2^{\beta \mathit{ve}}$ respectively of each edge $\mathit{uv}\in E(G)$ based on ve-degree of end vertices are defined as $$M_1^{\beta \mathit{ve}}(G)={\displaystyle \sum _{\mathit{uv}\in E}}({\mathrm{\Lambda }}_{\mathit{ve}}(u)+{\mathrm{\Lambda }}_{\mathit{ve}}(v)),$$ $$M_2^{\beta \mathit{ve}}(G)={\displaystyle \sum _{\mathit{uv}\in E}}({\mathrm{\Lambda }}_{\mathit{ve}}(u)\times {\mathrm{\Lambda }}_{\mathit{ve}}(v))\text{.}$$ The distribution of the sections has the following details, see the procedure detail in Fig. 8. Section 2 describe the schematic computer-based algorithm to calculate the degree vector, ve-degree vector, and eV-degree matrix and the respective topological indices for a given graph through Maple software. The Section 4 describes the quantitative-structure–property analysis (QSPR) of phytochemicals screened against SARS-CoV- $23{\mathit{CL}}^{\mathit{pro}}$ with the help of eV and ve degree-based topological descriptors. Applications to certain anticancer drug (Camptothecin-Polymer Conjugate IT-101) is presented in Section 5. Section 6 is the numerical results and discussion and Section 7 is conclusion. Section 3 describes an illustrative example to demonstrate the algorithm described in Section 2.

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Fig. 8

Working flowchart.

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2 Algorithm

a, b, c, d, e, f, g, h, k, l, m, n, o, p and q all are variables whose data type is an integer. We are using these variables for iterating through the matrix using indices in a loop.

3 Illustrative example on Algorithm

We calculated the degree vector, ve-degree vector, and eV-degree matrix and the respective topological indices for a given graph through the Maple algorithm given above. For this we consider the example of starphene graph (S) for $n=2,m=2$ and $l=2$ , see Fig. 9. First, we write the adjacency matrix (A) of the graph S in MAPLE. We get the adjacency matrix A by newGraph software (procedure mentioned in (Hayat and Khan, 2021)). As there are 18 vertices in the graph, the order of the A will be 18–18 and from figure it is clear that the graph S has 21 edges. $$A=\left[\begin{array}{cccccccccccccccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$$ Now, with the help of the adjacency matrix we can calculate the degree of vertices, Ve-degree and Ev-degree. The degree of a vertex can be obtained by adding the entries of the corresponding row and we get the degree sequence of starphene graph S in the form of row matrix B. $$B=[2,2,2,3,3,2,2,2,2,3,3,2,2,2,2,3,3,2]$$ Note that the degree sequence is according to the vertices labeled in Fig. 9. For Ve-degree of vertices, we multiplied the adjacency matrix A and B. The column matrix AB represent the Ve-degree of vertices. $$\mathit{AB}=[4,4,5,8,8,5,4,4,5,8,8,5,4,4,5,8,8,5]$$ Let $A_{\mathit{ij}}$ denote the entry in the i-th row and j-th column of adjacency matrix A and $B_i$ denoted the i-th entry of row matrix B. We multiply the entry $A_{\mathit{ij}}$ of adjacency matrix A with the entry $B_i+B_j$ to obtain a entry $L_{\mathit{ij}}$ of matrix L. The nonzero entries in the upper/lower triangular form of this matrix L give the Ev-degree list of the given graph. $$L=\left[\begin{array}{cccccccccccccccccc}\hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$$ We use separate formula commands against each index. The calculated values of the topological indices for the given graph are $$\begin{array}{cc}\hfill M^{\mathit{ev}}(S)=& 510\hfill \\ \hfill M_1^{\alpha \mathit{ve}}(S)=& 630\hfill \\ \hfill M_1^{\beta \mathit{ve}}(S)=& 252\hfill \\ \hfill M_2^{\beta \mathit{ve}}(S)=& 792\hfill \\ \hfill R^{\mathit{ev}}(S)=& \frac{3\sqrt{5}}{5}+\frac{3\sqrt{10}}{10}+\frac{3}{2}\hfill \\ \hfill \end{array}$$

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Fig. 9

Molecular structure of starphene for $n=2,m=2$ and $l=3$ .

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4 Quality testing analysis

In this section, we study the QSPR of phytochemicals that are used for screening against SARS- CoV- $23{\mathit{CL}}^{\mathit{pro}}$ using eV and ve degree-based topological descriptors. The productivity of the topological indices listed above was checked using the phytochemical data set contained in (Balaban, 1982) and https://pubchem.ncbi.nlm.nih.gov/. The data contains the information of the following variables: binding affinity, molecular weight, docking score and topological polar surface. IDs of phytochemicals are mentioned in Table 1 and their properties in Table 2. We could not found topological polar surface and molecular weight of NPACT00105. Therefore, we do not include NPACT00105 for QSPR. Ve-degree and eV-degree based topological indices are presented in Table 3. Table 4 shows the results of correlation coefficients between each pair of variables. We note that all the results are highly significant at 1% level of significance. This shows that there exists a strong positive linear bivariate relationship in the group of variables. The same extent of correlation is depicted in the Figs. (10) and (11). The table suggests that molecular weight (MW) and topological polar surface (TPS) depend on other indices such as $M^{\mathit{ev}},R^{\mathit{ev}},M_1^{\alpha \mathit{ve}},M_1^{\beta \mathit{ve}}$ and $M_2^{\beta \mathit{ve}}$ . Keeping in mind the above relationships, we consider the following linear regression model for prediction analyses. (see Fig. 12) $$P=a+\mathit{bTI}$$ Where P consists of either MW or TPS and TI denotes a predictor in the list ( $M^{\mathit{ev}},R^{\mathit{ev}},M_1^{\alpha \mathit{ve}},M_1^{\beta \mathit{ve}},M_2^{\beta \mathit{ve}}$ ).

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Fig. 10

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

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Fig. 11

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

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Fig. 12

Molecular Structure of IT-101. The components of the parentpolymer are polyethylene glycol and $\beta$ -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.

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Based on the data in Table 3 with 12 observations in each variable, we have the following estimated regression equations which can be used to predict molecular weight for a given value of any index $M^{\mathit{ev}},R^{\mathit{ev}},M_1^{\alpha \mathit{ve}},M_1^{\beta \mathit{ve}}$ and $M_2^{\beta \mathit{ve}}$ . $$\begin{array}{cc}\hfill \mathit{MW}=& a+\mathit{bX}\hfill \\ \hfill \mathit{MW}=& 189.809+0.159M^{\mathit{ev}}\hfill \\ \hfill \mathit{MW}=& 148.351+11.687R^{\mathit{ev}}\hfill \\ \hfill \mathit{MW}=& 198.166+0.133M_1^{\alpha \mathit{ve}}\hfill \\ \hfill \mathit{MW}=& 171.654+0.377M_1^{\beta \mathit{ve}}\hfill \\ \hfill \mathit{MW}=& 178.179+0.103M_2^{\beta \mathit{ve}}\hfill \\ \hfill \end{array}$$

Table 5 gives the results for correlation coefficient (r), coefficient of determination ( $R^2$ ), F-statistic (F) and standard error of the estimates (s) for simple linear regression models between MW and various predictors. The results exhibit that the estimated model of MW on $M_1^{\beta \mathit{ve}}$ consists of highest coefficient of determination and F-value ( $R^2$ = 0.9742, F = 375.812) and least standard error (s = 37.0609). This means that the model is highly significant. The only index $M_1^{\beta \mathit{ve}}$ has a high influence on MW and the observed values fall closer to the fitted line. Thus, $M_1^{\beta \mathit{ve}}$ is the best predictor of molecular weight. Similarly, $M^{\mathit{ev}}$ predicts molecular weight better than others after $M_1^{\beta \mathit{ve}}$ . A priority list of indices according to their performance is given in Table 7.

Following are the estimated models for predicting topological polar surface for a given value of any index $M^{\mathit{ev}},R^{\mathit{ev}},M_1^{\alpha \mathit{ve}},M_1^{\beta \mathit{ve}}$ and $M_2^{\beta \mathit{ve}}$ . The estimation results are based on the data in Table 3 with 12 observations in each variable. $$\begin{array}{cc}\hfill \mathit{TPS}=& 53.422+0.070M^{\mathit{ev}}\hfill \\ \hfill \mathit{TPS}=& 21.712+5.572R^{\mathit{ev}}\hfill \\ \hfill \mathit{TPS}=& 58.694+0.058M_1^{\alpha \mathit{ve}}\hfill \\ \hfill \mathit{TPS}=& 39.863+0.172M_1^{\beta \mathit{ve}}\hfill \\ \hfill \mathit{TPS}=& 47.565+0.046M_2^{\beta \mathit{ve}}\hfill \\ \hfill \end{array}$$

Table 6 gives the results of various statistics for bivariate regression equations of TPS on either of $M^{\mathit{ev}},R^{\mathit{ev}},M_1^{\alpha \mathit{ve}},M_1^{\beta \mathit{ve}}$ and $M_2^{\beta \mathit{ve}}$ . We observe that the estimated regression of TPS on $M_1^{\beta \mathit{ve}}$ possesses highest values of coefficient of determination and F-statistic ( $R^2$ = 0.7850, F = 36.529) and least value of standard error of regression (s = 54.4280). The results reveal that the regression model (TPS on $M_1^{\beta \mathit{ve}}$ ) is highly significant. TPS is highly determined by only $M_1^{\beta \mathit{ve}}$ and the estimated line fits the observations closer than other lines. Hence, $M_1^{\beta \mathit{ve}}$ seems to the best predictor of topological polar surface as well. The hierarchical order of the indices according to their performance is shown in Table 7.

Based on correlation and regression analyses, we may conclude that both molecular weight and the topological polar surface can be predicted well by the indices in the order mentioned in Table 7. However, these indices cannot determine the other variables well such as docking scores (DS) and binding affinity (BA). Table 8 shows the results of the bivariate correlation coefficient which are all insignificant. This means that DS and BA are not linearly correlated with all the indices $M^{\mathit{ev}},R^{\mathit{ev}}$ , $M_1^{\alpha \mathit{ve}},M_1^{\beta \mathit{ve}}$ and $M_2^{\beta \mathit{ve}}$ . Thus, linear regression cannot model their relationship.

5 Camptothecin-Polymer Conjugate IT-101

Camptothecin (CA) is an alkaloid with a unique anticancer function, operates with a vital and unique mechanism of the action to target the topoisomerase (I) nuclear enzyme. The development of a large range of tumors is inhibited by CA. IT-101 is a conjugate of camptothecin with a cyclodextrin-based polymer (Schluep et al., 2006). Topoisomerase I is affected by a drug known as Camptothecin, derived from the Chinese tree, that allows the DNA cleavage, but it inhibits the subsequent ligation and breaks the DNA chain. The systemic application of the camptothecin has some limitations due to severe toxicity. Camptothecin shows potent in vitro anti-glioma activity, making it another common polymer delivery candidate. In the 9L gliosarcoma rat model, sodium salt of camptothecin loaded into 50 percent polymers was checked and resulted in a substantial survival increase (Thomas et al., 2004). The median survival with camptothecin polymers was nineteen days in the model of control rats and up to 120 days. Compared with controls, when we used direct intratumoral injection of our drug camptothecin without polymer then it showed no betterment in the survival (Liu et al., 2000). Studies integrating camptothecin into 50% polymers for the treatment of 9L gliosarcoma in the rats. It gives protection and efficacy with the survival of 69 days in animals that were treated with the drug camptothecin. Under similar conditions, polymer administration of intracranial camptothecin showed a substantial betterment in survival as compared with the alone camptothecin (Schultz, 1973). In our work, we compute five ve-degree and eV-degree based topological indices (Chen et al., 2021) of Camptothecin-Polymer Conjugate IT-101. The vertex set of Camptothecin-Polymer Conjugate IT-101 can be partitioned in four set based on the degree of vertices. There are $4\mathit{mn}+129m+7$ vertices of degree $1,\mathit{mn}+33m$ vertices of degree $2,43m+1$ vertices of degree 3 and $2\mathit{mn}+62m+2$ vertices of degree 4. Similarly, the Edge partition $E_{(i,j)}$ of IT-101 based on the degree of end vertices i and j of an edge $e=\mathit{ij}$ is as follows; $E_{(1,2)}$ with $19m$ edges, $E_{(1,3)}$ with $25m+1$ edges, $E_{(1,4)}$ with $4\mathit{mn}+85m+6$ edges, $E_{(2,3)}$ with $8m$ edges, $E_{(2,4)}$ with $\mathit{mn}+60m$ edges, $E_{(3,3)}$ with $35m$ edges, $E_{(3,4)}$ with $20m+2$ edges and $E_{(4,4)}$ with $\mathit{mn}+43m$ edges.

(a) The Zagreb index $$\begin{array}{cc}\hfill M^{\mathit{ev}}(H)=& {\displaystyle \sum _{e\in E(H)}}{\mathrm{\Lambda }}_{\mathit{ev}}{(e)}^2,\hfill \\ \hfill M^{\mathit{ev}}(H)=& {(3)}^2|E_{(1,2)}|+{(4)}^2|E_{(1,3)}|+{(5)}^2|E_{(1,4)}|+{(5)}^2|E_{(2,3)}|+{(6)}^2|E_{(2,4)}|\hfill \\ \hfill +& {(6)}^2|E_{(3,3)}|+{(7)}^2|E_{(3,4)}|+{(8)}^2|E_{(4,4)}|\hfill \\ \hfill =& 200\mathit{mn}+10048m+264\text{.}\hfill \end{array}$$

(b) The Randic index $$\begin{array}{cc}\hfill R^{\mathit{ev}}(H)=& {\displaystyle \sum _{e\in E(H)}}{\mathrm{\Lambda }}_{\mathit{ev}}{(e)}^{-\frac{1}{2}},\hfill \\ \hfill R^{\mathit{ev}}(H)=& {(3)}^{-\frac{1}{2}}|E_{(1,2)}|+{(4)}^{-\frac{1}{2}}|E_{(1,3)}|+{(5)}^{-\frac{1}{2}}|E_{(1,4)}|+{(5)}^{-\frac{1}{2}}|E_{(2,3)}|+{(6)}^{-\frac{1}{2}}|E_{(2,4)}|\hfill \\ \hfill +& {(6)}^{-\frac{1}{2}}|E_{(3,3)}|+{(7)}^{-\frac{1}{2}}|E_{(3,4)}|+{(8)}^{-\frac{1}{2}}|E_{(4,4)}|\hfill \\ \hfill =& (\frac{4}{\sqrt{5}}+\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{8}})\mathit{mn}+(\frac{19}{\sqrt{3}}+\frac{25}{\sqrt{4}}+\frac{85}{\sqrt{5}}+\frac{8}{\sqrt{5}}+\frac{60}{\sqrt{6}}+\frac{35}{\sqrt{6}}+\frac{20}{\sqrt{7}}+\frac{43}{\sqrt{8}})m\hfill \\ \hfill +& (\frac{1}{\sqrt{4}}+\frac{6}{\sqrt{5}}+\frac{2}{\sqrt{7}})\text{.}\hfill \end{array}$$

By using the above table, we have a first ve-degree based Zagreb $\alpha$ -index. $$\begin{array}{cc}\hfill M_1^{\alpha \mathit{ve}}(H)=& {\displaystyle \sum _{v\in V(H)}}{\mathrm{\Lambda }}_{\mathit{ve}}{(v)}^2\hfill \\ \hfill M_1^{\alpha \mathit{ve}}(H)=& {(2)}^2(19m)+{(3)}^2(25m+1)+{(4)}^2(4\mathit{mn}+85m+6)+{(5)}^2(19m)+{(6)}^2(2m)+{(7)}^2(4m)\hfill \\ \hfill +& +{(8)}^2(\mathit{mn}+8m)+{(7)}^2(18m)+{(8)}^2(12m)+{(9)}^2(5m+1)+{(10)}^2(8m)\hfill \\ \hfill +& {(6)}^2(2)+{(7)}^2(4m)+{(8)}^2(2\mathit{mn}+15m)+{(9)}^2(9m)+{(10)}^2(2m)\hfill \\ \hfill +& {(11)}^2(30m)+{(12)}^2(2m)\hfill \\ \hfill =& 256\mathit{mn}+11774m+258\text{.}\hfill \end{array}$$

By using the table we can compute indices based on ve-degree of end vertices of each edge as (a) The first Zagreb $\beta$ -index $$\begin{array}{cc}\hfill M_1^{\beta \mathit{ve}}(H)=& {\displaystyle \sum _{\mathit{uv}\in E(H)}}({\mathrm{\Lambda }}_{\mathit{ve}}(u)+{\mathrm{\Lambda }}_{\mathit{ve}}(v))\hfill \\ \hfill M_1^{\beta \mathit{ve}}(H)=& (7)|E_1^{\ast }|+(10)|E_2^{\ast }|+(11)|E_3^{\ast }|+(12)|E_4^{\ast }|+(10)|E_5^{\ast }|+(11)|E_6^{\ast }|+(12)|E_7^{\ast }|\hfill \\ \hfill +& (13)|E_8^{\ast }|+(14)|E_9^{\ast }|+(15)|E_{10}^{\ast }|+(14)|E_{11}^{\ast }|+(14)|E_{12}^{\ast }|+(13)|E_{13}^{\ast }|+(16)|E_{14}^{\ast }|\hfill \\ \hfill +& (14)|E_{15}^{\ast }|+(19)|E_{16}^{\ast }|+(16)|E_{17}^{\ast }|+(17)|E_{18}^{\ast }|+(19)|E_{19}^{\ast }|+(14)|E_{20}^{\ast }|+(15)|E_{21}^{\ast }|\hfill \\ \hfill +& (16)|E_{22}^{\ast }|+(17)|E_{23}^{\ast }|+(16)|E_{24}^{\ast }|+(17)|E_{25}^{\ast }|+(18)|E_{26}^{\ast }|+(19)|E_{27}^{\ast }|+(20)|E_{28}^{\ast }|\hfill \\ \hfill +& (15)|E_{29}^{\ast }|+(19)|E_{30}^{\ast }|+(16)|E_{31}^{\ast }|+(17)|E_{32}^{\ast }|+(18)|E_{33}^{\ast }|+(19)|E_{34}^{\ast }|+(15)|E_{35}^{\ast }|\hfill \\ \hfill +& (20)|E_{36}^{\ast }|+(17)|E_{37}^{\ast }|+(22)|E_{38}^{\ast }|+(17)|E_{39}^{\ast }|+(16)|E_{40}^{\ast }|+(17)|E_{41}^{\ast }|+(19)|E_{42}^{\ast }|\hfill \\ \hfill +& (22)|E_{43}^{\ast }|+(22)|E_{44}^{\ast }|+(20)|E_{45}^{\ast }|+(18)|E_{46}^{\ast }|\hfill \\ \hfill =& (7)(19m)+(10)(18m)+(11)(6m)+(12)(m+1)+(10)(6)+(11)(10m)\hfill \\ \hfill +& (12)(4\mathit{mn}+30m)+(13)(11m)+(14)(4m)+(15)(30m)+(14)(4m)+(14)(4m)\hfill \\ \hfill +& (13)(5m)+(16)(14m)+(14)(4m)+(19)(2m)+(16)(\mathit{mn}+7m)+(17)(14m)\hfill \\ \hfill +& (19)(14m)+(14)(6m)+(15)(2m)+(16)(6m)+(17)(8m)+(16)(3m)+(17)(4m)\hfill \\ \hfill +& (18)(2m)+(19)(2m)+(20)(2m)+(15)(2m)+(19)(2m)+(16)(2m)+(17)(m)\hfill \\ \hfill +& (18)(4m)+(19)(3m)+(15)(m+1)+(20)(m)+(17)(2m)+(22)(2m)+(17)(2m)\hfill \\ \hfill +& (16)(\mathit{mn})+(17)(2m)+(19)(9m)+(22)(2m)+(22)(21m)+(20)(7m)+(18)(1)\hfill \\ \hfill =& 80\mathit{mn}+4441m+105\text{.}\hfill \end{array}$$

(b) The second Zagreb $\beta$ -index $$\begin{array}{cc}\hfill M_2^{\beta \mathit{ve}}(H)=& {\displaystyle \sum _{\mathit{uv}\in E(H)}}({\mathrm{\Lambda }}_{\mathit{ve}}(u)\times {\mathrm{\Lambda }}_{\mathit{ve}}(v))\hfill \\ \hfill M_2^{\beta \mathit{ve}}(H)=& (10)|E_1^{\ast }|+(21)|E_2^{\ast }|+(24)|E_3^{\ast }|+(27)|E_4^{\ast }|+(24)|E_5^{\ast }|+(28)|E_6^{\ast }|+(32)|E_7^{\ast }|\hfill \\ \hfill +& (36)|E_8^{\ast }|+(40)|E_9^{\ast }|+(44)|E_{10}^{\ast }|+(48)|E_{11}^{\ast }|+(49)|E_{12}^{\ast }|+(40)|E_{13}^{\ast }|+(55)|E_{14}^{\ast }|\hfill \\ \hfill +& (49)|E_{15}^{\ast }|+(84)|E_{16}^{\ast }|+(64)|E_{17}^{\ast }|+(72)|E_{18}^{\ast }|+(88)|E_{19}^{\ast }|+(49)|E_{20}^{\ast }|+(56)|E_{21}^{\ast }|\hfill \\ \hfill +& (63)|E_{22}^{\ast }|+(70)|E_{23}^{\ast }|+(64)|E_{24}^{\ast }|+(72)|E_{25}^{\ast }|+(80)|E_{26}^{\ast }|+(90)|E_{27}^{\ast }|+(100)|E_{28}^{\ast }|\hfill \\ \hfill +& (56)|E_{29}^{\ast }|+(84)|E_{30}^{\ast }|+(64)|E_{31}^{\ast }|+(72)|E_{32}^{\ast }|+(80)|E_{33}^{\ast }|+(88)|E_{34}^{\ast }|+(54)|E_{35}^{\ast }|\hfill \\ \hfill +& (99)|E_{36}^{\ast }|+(70)|E_{37}^{\ast }|+(120)|E_{38}^{\ast }|+(70)|E_{39}^{\ast }|+(64)|E_{40}^{\ast }|+(72)|E_{41}^{\ast }|+(88)|E_{42}^{\ast }|\hfill \\ \hfill +& (120)|E_{43}^{\ast }|+(121)|E_{44}^{\ast }|+(99)|E_{45}^{\ast }|+(81)|E_{46}^{\ast }|\hfill \\ \hfill =& (10)(19m)+(21)(18m)+(24)(6m)+(27)(m+1)+(24)(6)+(28)(10m)\hfill \\ \hfill +& (32)(4\mathit{mn}+30m)+(36)(11m)+(40)(4m)+(44)(30m)+(48)(4m)+(49)(4m)\hfill \\ \hfill +& (40)(5m)+(55)(14m)+(49)(4m)+(84)(2m)+(64)(\mathit{mn}+7m)+(72)(14m)\hfill \\ \hfill +& (88)(7m)+(49)(6m)+(56)(2m)+(63)(6m)+(70)(8m)+(64)(3m)+(72)(4m)\hfill \\ \hfill +& (80)(2m)+(90)(2m)+(100)(2m)+(56)(2m)+(84)(2m)+(64)(2m)+(72)(m)\hfill \\ \hfill +& (80)(4m)+(88)(3m)+(54)(m+1)+(99)(m)+(70)(2m)+(120)(2m)\hfill \\ \hfill +& (70)(2m)+(64)(\mathit{mn})+(72)(2m)+(88)(9m)+(120)(2m)+(121)(21m)\hfill \\ \hfill +& (99)(7m)+(81)(1)\hfill \\ \hfill =& 256\mathit{mn}+16160m+306\text{.}\hfill \end{array}$$