Distance based and bond additive topological indices of certain repurposed antiviral drug compounds tested for treating COVID‐19

Abstract

The entire world is struggling to control the spread of coronavirus (COVID‐19) as there are no proper drugs for treating the disease. Under clinical trials, some of the repurposed antiviral drugs have been applied to COVID‐19 patients and reported the efficacy of the drugs with the diverse inferences. Molecular topology has been developed in recent years as an influential approach for drug design and discovery in which molecules that are structurally related show similar pharmacological properties. It permits a purely mathematical description of the molecular structure so that in the development of identification of new drugs can be found through adequate topological indices. In this paper, we study the structural properties of the several antiviral drugs such as chloroquine, hydroxychloroquine, lopinavir, ritonavir, remdesivir, theaflavin, nafamostat, camostat, umifenovir and bevacizumab by considering the distance and bond measures of chemical compounds. Our quantitative values of the topological indices are extremely useful in the recent development of designing new drugs for COVID‐19.

The primary results of the paper include distance‐based and bond additive topological descriptors of antiviral medications for the treatment of COVID‐19 like chloroquine, hydroxychloroquine, lopinavir, ritonavir, remdesivir, theaflavin, nafamostat, camostat, umifenovir, and bevacizumab by graph‐theoretic technique, in particular, strength‐weighted quotient graph.

1. INTRODUCTION

Chemical graph theory is an interdisciplinary science that applies the graph theory in the study of the topology of a chemical compound. The analysis of a molecular formula to determine the relationship between the structure and the property which has been used in the field of QSAR/QSPR studies [1, 2]. There are numerous methods to study the molecular structures, in which many chemists favor to use the topological indices as molecular descriptors to calculate toxicity and predict biological activity. The topological indices are numerical quantities with many important applications in structural chemistry.

The expansion of the group of diseased and the increasingly more developed medication innovation, stimulate the advancement of recently invented drugs every year. Consequently, it requires an enormous amount of work to decide the pharmacological, chemical and biological activities of these new drugs and such remaining tasks at hand become increasingly difficult and clustered. Moreover, it needs enough reagents equipment and researchers to validate the experiments and the reactions of existing new drugs [3]. Subsequently, it is ended up being proficient to clarify the medicinal properties and it can ensure the role of the drugs in treating the disease. Hence, the strategies on topological index calculation can assist with getting the accessible biological and medical data on new drugs without much help of in vitro and in vivo studies [4, 5]. The theory of topological index came from the work done by Wiener while he was functioning on the paraffin boiling points using path numbers and polarity indices (see Klavžar and Nadjafi‐Arani [6] for details).

Coronavirus (CoV) belongs to a subfamily of large and enveloped single RNA virus. It causes simple fever which leads to progressively serious diseases, for example, Middle East respiratory syndrome (MERS‐CoV) and severe acute respiratory syndrome (SARS‐CoV). There are four types of coronaviruses namely alpha, beta, gamma, and delta in which alpha and beta viruses can infect mammals. The current novel coronavirus (2019‐nCoV) belongs to the beta coronaviruses which emerged from in Wuhan City, China [7]. The terrible, deadly disease epidemic COVID‐19 around the world and unfortunately, there are no specific drugs for the disease that are presently available and therefore, it is critical to find suitable antiviral agents from the existing drugs for the treatment of COVID‐19. In the literature, several antiviral drugs such as chloroquine, hydroxychloroquine, lopinavir, ritonavir, remdesivir, theaflavin, nafamostat, camostat, umifenovir and bevacizumab are investigated to inhibit the infection and transmission of the 2019‐nCoV in vitro and thereby obtained the optimistic results [8, 9, 10, 11, 12, 13].

Chloroquine is a proven anti‐malarial drug and known that the anti‐viral and anti‐inflammatory activities led the way in the treatment of COVID‐19. It inhibits the enzyme heme polymerase that changes the toxic heme into non‐toxic hemazoin, which plays a major role in malarial infection. The drug is prescribed for pneumonia caused by COVID‐19 [14]. In Colson et al. [15], the authors found that chloroquine and hydroxychloroquine as available weapons to fight COVID‐19. Hydroxychloroquine [16] is a plagiaristic of chloroquine that has both anti‐inflammatory and anti‐malarial activities and is now most often used as an anti‐rheumatologic agent in systemic lupus erythematosis and rheumatoid arthritis. The molecular compounds of chloroquine and hydroxychloroquine are depicted in Figure 1.

FIGURE 1.

(A) Chloroquine C ₁₈ H ₂₆ ClN ₃ , (B) hydroxychloroquine C ₁₈ H ₂₆ ClN ₃ O

Lopinavir and ritonavir belong to a class protease inhibitors. Lopinavir is broadly utilized for the treatment of HIV and is a likely possibility for the treatment of COVID‐19. Ritonavir inhibits the metabolizing enzyme cytochrome P450 3A and consequently increases the half‐life of lopinavir. The combination is used with other medications to treat the human immunodeficiency virus infection [12]. Remdesivir is a phosphoramidate prodrug with potential antiviral activity against a variety of RNA viruses. It has been used in the treatment of Ebola virus infection and currently considered a potential drug [13]. Theaflavins are polymers derived from natural catechins which are oxidized upon drying of the plant leaves. The chemical compounds of these drugs are shown in the Figures 2 and 3.

FIGURE 2.

(A) Lopinavir C ₃₇ H ₄₈ N ₄ O ₅ , (B) ritnovir C ₃₇ H ₄₈ N ₆ O ₅ S ₂

FIGURE 3.

(A) Remdesivir C ₂₇ H ₃₅ N ₆ O ₈ P, (B) theaflavin C ₂₉ H ₂₄ O ₁₂

Nafamostat and camostat are serine protease inhibitors which are authorized in Japan and South Korea for the treatment of pancreatitis in humans. They are proven and well‐studied drugs [17, 18]. Umifenovir is a hydrophobic molecule which blocks/inhibits the entry of virus into the cell. These outcomes in a perfect mix for successfully handling high popular burdens in COVID‐19 patients during the beginning phase of the infection [12]. Vascular endothelial growth factor (VEGF) is considered as the most intense vascular penetrability inducers. Ongoing proof has uncovered that higher VEGF levels in COVID‐19 patients contrasted and sound controls. Various examinations have affirmed a key job of VEGF as expected remedial objective in intense lung injury and respiratory pain disorder. It was proved that bevacizumab is an enemy of VEGF prescription and may offer an extraordinary way to deal with treat with COVID‐19 [19]. The molecular compounds of these drugs are given in the Figures 4 and 5.

FIGURE 4.

(A) Nafamostat C ₁₉ H ₁₇ N ₅ O ₂ , (B) camostat C ₂₀ H ₂₂ N ₄ O ₅

FIGURE 5.

(A) Umifenovir C ₂₂ H ₂₅ BrN ₂ O ₃ S, (B) bevacizumab C ₂₂ H ₂₄ N ₄ OS

In the recent paper [5] several degree based topological indices of chloroquine, hydroxychloroquine, remdesivir and theaflavin compounds were computed. In this study, we obtain the distance based and bond additive topological indices such as Wiener, edge‐Wiener, Wiener polarity, Szeged, edge‐Szeged, PI, Mostar and edge‐Mostar for the above described 10 drug compounds.

2. GRAPH THEORETICAL CONCEPTS

Let G = (V(G), E(G)) be a simple graph. The degree of a vertex v is denoted as d _G (v) which is characterized as the number of edges incident to v . The number of pentagons and hexagons of G are denoted by N _p(G) and N _h(G) respectively. For any positive integer i , we represent $N_G^i\left(v\right)=\left\{u\in V\left(G\right):d_G\left(u,v\right)=i\right\}$ as the i ^th neighborhood of v , and thus clearly the open neighborhood of v (denoted by N _G (v)) is $N_G^1\left(v\right)$. The distance between a vertex v ∈ V(G) and an edge e = ab ∈ E(G), denoted by d _G (v, e), is defined as min{d _G (v, a), d _G (v, b)}. The distance between two edges e = ab and f = cd of G is defined as the minimum number of edges along a shortest (e, c)‐path or a shortest (e, d)‐path and denoted by D _G (e, f). For an edge e = uv ∈ E(G), we characterize the following accompanying sets for the end vertex u :

$$\begin{array}{l}{N}_u\left(e|G\right)=\left\{x\in V\left(G\right):d_G\left(u,x\right)<d_G\left(v,x\right)\right\},\\ M_u\left(e|G\right)=\left\{y\in E\left(G\right):d_G\left(u,y\right)<d_G\left(v,y\right)\right\}.\end{array}$$

In fact, N _u (e| G) and M _u (e| G) are the set of all vertices and edges of G which are nearer to u than to v whose cardinality is defined as n _u (e| G) and m _u (e| G) respectively. The values n _v (e| G) and m _v (e| G) are characterized similarly.

The strength‐weighted graph was at first presented in Arockiaraj et al. [20] as G _sw = (G, [w _v, s _v], s _e) where w _v is the vertex weight, s _v is the vertex strength and s _e is the edge strength. The distance between any two vertices in G _sw is denoted as $d_{G_{\mathrm{sw}}}\left(u,v\right)=d_G\left(u,v\right)$. Essentially, the sets N _u (e| G _sw) = N _u (e| G) and M _u (e| G _sw) = M _u (e| G) are described with cardinality n _u (e| G _sw) = $\sum _{x\in N_u\left(e|G_{\mathrm{sw}}\right)}{w}_{\mathrm{v}}\left(x\right)$ and m _u (e| G _sw) = $\sum _{x\in N_u\left(e|G_{\mathrm{sw}}\right)}{s}_{\mathrm{v}}\left(x\right)+\sum _{y\in M_u\left(e|G_{\mathrm{sw}}\right)}{s}_{\mathrm{e}}\left(y\right)$. The values of n _v(e| G _sw) and m _v(e| G _sw) are analogous. Several topological indices (TI) for the strength‐weighted graphs were studied in References [20, 21, 22, 23, 24] and shown in Table 1. It is to be noted that TI(G) = TI(G _sw) when w _v = 1, s _v = 0, and s _e = 1.

TABLE 1.

Topological indices of strength‐weighted graph G _sw

Topological indices	Mathematical expressions
Wiener	$W\left(G_{\mathrm{sw}}\right)=\sum _{\left\{u,v\right\}\subseteq V\left(G_{\mathrm{sw}}\right)}{w}_{\mathrm{v}}\left(u\right)w_{\mathrm{v}}\left(v\right)d_{G_{\mathrm{sw}}}\left(u,v\right)$
Edge‐Wiener	$\begin{array}{l}{W}_{\mathrm{e}}\left(G_{\mathrm{sw}}\right)=\sum _{\left\{u,v\right\}\subseteq V\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{v}}\left(u\right)s_{\mathrm{v}}\left(v\right)d_{G_{\mathrm{sw}}}\left(u,v\right)\\ +\sum _{\left\{e,f\right\}\subseteq E\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{e}}\left(e\right)s_{\mathrm{e}}\left(f\right)D_{G_{\mathrm{sw}}}\left(e,f\right)\\ +\sum _{u\in V\left(G_{\mathrm{sw}}\right)}\sum _{f\in E\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{v}}\left(u\right)s_{\mathrm{e}}\left(f\right)d_{G_{\mathrm{sw}}}\left(u,f\right)\end{array}$
Szeged	${\mathit{Sz}}_{\mathrm{v}}\left(G_{\mathrm{sw}}\right)=\sum _{e=\mathit{uv}\in E\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{e}}\left(e\right)n_u\left(e|G_{\mathrm{sw}}\right)n_{\mathrm{v}}\left(e|G_{\mathrm{sw}}\right)$
Edge‐Szeged	${\mathit{Sz}}_{\mathrm{e}}\left(G_{\mathrm{sw}}\right)=\sum _{e=\mathit{uv}\in E\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{e}}\left(e\right)m_u\left(e|G_{\mathrm{sw}}\right)m_{\mathrm{v}}\left(e|G_{\mathrm{sw}}\right)$
Padmakar–Ivan	$\mathit{PI}\left(G_{\mathrm{sw}}\right)=\sum _{e=\mathit{uv}\in E\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{e}}\left(e\right)\left[m_u\left(e|G_{\mathrm{sw}}\right)+m_{\mathrm{v}}\left(e|G_{\mathrm{sw}}\right)\right]$
Mostar	$\mathit{Mo}\left(G_{\mathrm{sw}}\right)=\sum _{e=\mathit{uv}\in E\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{e}}\left(e\right)\mid n_u\left(e|G_{\mathrm{sw}}\right)-n_{\mathrm{v}}\left(e|G_{\mathrm{sw}}\right)\mid$
Edge‐Mostar	${\mathit{Mo}}_{\mathrm{e}}\left(G_{\mathrm{sw}}\right)=\sum _{e=\mathit{uv}\in E\left(G_{\mathrm{sw}}\right)}{s}_{\mathrm{e}}\left(e\right)\mid m_u\left(e|G_{\mathrm{sw}}\right)-m_{\mathrm{v}}\left(e|G_{\mathrm{sw}}\right)\mid$

The cut method ended up being incredibly convenient when managing a distance‐based graph invariants which are thusly among the focal ideas of molecular graph theory [6, 25, 26]. Let us recollect the concepts of isometric subgraph, partial cubes, convex subgraph and Djoković‐Winkler Θ condition which are the key documentations of the cut technique. A graph H is supposed to be an isometric subgraph of a graph G , if for u, v ∈ V(H), d _G (u, v) = d _H (u, v). The very much characterized assortment of such subgraphs of hypercubes are called partial cubes.

For any two vertices, the shortest paths between them lies inside the same subgraph then the subgraph is said to be a convex subgraph and the condition, d _G (s ₁, s ₂) + d _G (t ₁, t ₂) ≠ d _G (s ₁, t ₂) + d _G (t ₁, s ₂) for two edges e ₁ = s ₁ t ₁ and e ₂ = s ₂ t ₂ is called Djoković‐Winkler (Θ) relation. This relation Θ is always reflexive, symmetric and transitive in case of partial cubes but not transitive in general. Hence the Θ partitions of the edge set of a partial cube G into classes F ₁, F ₂, …F _r , called Θ‐classes or convex cuts. However, its transitive closure Θ^* forms an equivalence relation in general and partitions the edge set into many convex components. A partition ℰ = {E ₁, E ₂, …E _k } of E(G) is said to be coarser than partition ℱ if each set E _i is the union of one or more Θ^* ‐classes of G . For any class E _i , the quotient graph G/E _i is formed from the disconnected graph G − E _i , where the connected components act as the vertices and the two components $C_j^i$ and $C_k^i$ are adjacent whenever a vertex $x\in C_j^i$ is adjacent to a vertex $y\in C_k^i$ with xy ∈ E _i .

Theorem 1

[20, 21, 22] For a strength‐weighted graph G _sw = (G, [w _v, s _v], s _e), let ℰ = {E ₁, E ₂, …E _k } be a partition of E(G) coarser than ℱ. Let TI represent the various topological indices such as W , W _e , Sz _v , Sz _e , PI , Mo , Mo _e . Then

$$\mathit{TI}\left(G_{\mathrm{sw}}\right)=\sum _{i=1}^k\mathit{TI}\left(G/E_i,\left[w_{\mathrm{v}}^i,s_{\mathrm{v}}^i\right],s_{\mathrm{e}}^i\right)$$

where

• $w_{\mathrm{v}}^i:V\left(G/E_i\right)\to {\mathrm{ℝ}}_0^{+}$ is defined by $w_{\mathrm{v}}^i\left(C\right)=\sum _{x\in V\left(C\right)}{w}_{\mathrm{v}}\left(x\right)$, for all connected components C ∈ G/E _i ,

• $s_{\mathrm{v}}^i:V\left(G/E_i\right)\to {\mathrm{ℝ}}_0^{+}$ is defined by $s_{\mathrm{v}}^i\left(C\right)=\sum _{\mathit{xy}\in E\left(C\right)}{s}_{\mathrm{e}}\left(\mathit{xy}\right)+\sum _{x\in V\left(C\right)}{s}_{\mathrm{v}}\left(x\right)$, for all connected components C ∈ G/E _i ,

• $s_{\mathrm{e}}^i:E\left(G/E_i\right)\to {\mathrm{ℝ}}_0^{+}$ is defined as the number of edges in E _i such that one end in C and the other end in D , for any two connected components C and D of G/E _i .

3. DISTANCE BASED TOPOLOGICAL INDICES

The aim of this section is to compute several distance‐based and bond additive topological indices of drug compounds considered in our study. In the sequel, we assume that the vertex strength‐weighted values of a quotient graph for the convex cut F _i of G as [a _i , b _i ], [c _i , d _i ] and edge strength as e _i such that c _i =  ∣ V(G) ∣  − a _i and d _i =  ∣ E(G) ∣  − b _i  − e _i .

Theorem 2

Let G ₁ be a chloroquine compound. Then, W(G ₁) = 1212, W _e(G ₁) = 962, Sz _v(G ₁) = 1717, Sz _e(G ₁) = 1362, PI(G ₁) = 492, Mo(G ₁) = 284 and Mo _e(G ₁) = 312.

The chloroquine has 22 veritces and 23 edges. Let {F _i  : 1 ≤ i ≤ 17} be the various convex cuts of G ₁ and the corresponding quotient graph which are depicted in Figure 6. The strength‐weighted values G ₁/F _i are presented in Table 2 and in addition, e _i = 1, 1 ≤ i ≤ 12; e _i = 2, 13 ≤ i ≤ 16; e ₁₇ = 3.

FIGURE 6.

(A) Various cuts of chloroquine G ₁ , (B) quotient graph G ₁/F _i

TABLE 2.

Strength‐weighted values of quotient graph G ₁/F _i

F i	1 ≤ i ≤ 4	5	6	7	8	9	10	11	12	13	14	15	16	17
a i	1	2	5	2	6	7	8	10	11	4	4	3	8	6
b i	0	1	4	1	5	6	7	9	12	3	3	2	8	5

We now compute the topological indices as $W\left(G_1\right)={\sum }_{i=1}^{17}{a}_i{c}_i=1212$, $W_{\mathrm{e}}\left(G_1\right)={\sum }_{i=1}^{17}{b}_i{d}_i=962$, ${\mathit{Sz}}_{\mathrm{v}}\left(G_1\right)={\sum }_{i=1}^{17}{e}_i{a}_i{c}_i=1717$, ${\mathit{Sz}}_{\mathrm{e}}\left(G_1\right)={\sum }_{i=1}^{17}{e}_i{b}_i{d}_i=1362$, $\mathit{PI}\left(G_1\right)={\sum }_{i=1}^{17}{e}_i\left(b_i+d_i\right)=492$, $\mathit{Mo}\left(G_1\right)={\sum }_{i=1}^{17}{e}_i\mid a_i-c_i\mid =284$, and ${\mathit{Mo}}_{\mathrm{e}}\left(G_1\right)={\sum }_{i=1}^{17}{e}_i\mid b_i-d_i\mid =312$.

Theorem 3

Let G ₂ be a hydroxychloroquine compound. Then, W(G ₂) = 1390, W _e(G ₂) = 1118, Sz _v(G ₂) = 1926, Sz _e(G ₂) = 1544, PI(G ₂) = 538, Mo(G ₂) = 314 and Mo _e(G ₂) = 342.

The number of vertices and edges in hydroxychloroquine are 23 and 24 respectively. Let {F _i  : 1 ≤ i ≤ 18} be the various convex cuts of G ₂ as depicted in Figure 7. The strength‐weighted values of the quotient graph G ₂/F _i are given in Table 3 and e _i = 1, 1 ≤ i ≤ 13; e _i = 2, 14 ≤ i ≤ 17; e ₁₈ = 3. By simple mathematical calculation based on the cut method, we obtain the required results.

Theorem 4

Let G ₃ be a lopinovir compound. Then, W(G ₃) = 8131, W _e(G ₃) = 7662, Sz _v(G ₃) = 9796, Sz _e(G ₃) = 8868, PI(G ₃) = 2328, Mo(G ₃) = 1698 and Mo _e(G ₃) = 1832.

We have ∣V(G ₃) ∣  = 46, ∣E(G ₃) ∣  = 49 and there are 37 convex cuts for lopinovir compound, denoted by {F _i  : 1 ≤ i ≤ 37}, as depicted in Figure 8.

FIGURE 7.

(A) Various cuts of hydroxychloroquine G ₂ ; (B) quotient graph G ₂/F _i

TABLE 3.

Strength‐weighted values of quotient graph G ₂/F _i

F i	1 ≤ i ≤ 4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
a i	1	2	2	3	6	7	8	9	11	12	4	4	3	8	6
b i	0	1	1	2	5	6	7	8	10	11	3	3	2	8	5

FIGURE 8.

(A) Various cuts of lopinovir G ₃ , (B) G ₃/F _i

The strength‐weighted values of the quotient graph G ₃/F _i are given in Table 4 and e _i = 1, 1 ≤ i ≤ 25; e _i = 2, 26 ≤ i ≤ 37. We complete the proof by routine mathematical calculations.

Theorem 5

Let G ₄ be a ritnovir compound. Then, W(G ₄) = 10595, W _e(G ₄) = 9662, Sz _v(G ₄) = 11325, Sz _e(G ₄) = 10788, PI(G ₄) = 2552, Mo(G ₄) = 1841 and Mo _e(G ₄) = 1982.

Since ritnovir compound has two pentagons, we can conclude that it does not belong to the family of partial cubes. In this case, we use the Θ^* ‐partition to compute the required topological indices. Let {F _i  : 1 ≤ i ≤ 39} ∪ {S ₁, S ₂} be a such partition of G ₄ as depicted in Figure 9. The strength‐weighted values of the quotient graph G ₄/F _i are given in Table 5 and e _i = 1, 1 ≤ i ≤ 36; e _i = 2, 37 ≤ i ≤ 39. But the strength‐weighted values of the quotient graph G ₄/S _k , k = 1, 2 as given Figure 9 are (x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅) = (46,1,1,1,1); (x ₂₁, x ₂₂, x ₂₃, x ₂₄, x ₂₅) = (43,1,4,1,1); (y ₁₁, y ₁₂, y ₁₃, y ₁₄, y ₁₅) = (47,0,0,0,0); (y ₂₁, y ₂₂, y ₂₃, y ₂₄, y ₂₅) = (44,0,3,0,0), (e _k1, e _k2, e _k3, e _k4, e _k5) = (1,1,1,1,1). We have ∣V(G ₄) ∣  = 50, ∣E(G ₄) ∣  = 52 and now compute topological indices as follows:

$$\begin{array}{l}W\left(G_4\right)=\sum _{i=1}^{39}{a}_i{c}_i+\sum _{k=1}^2[x_{k1}\left(x_{k2}+x_{k5}\right)+x_{k2}{x}_{k3}+x_{k3}{x}_{k4}+x_{k4}{x}_{k5}+2(x_{k1}{x}_{k3}+x_{k1}{x}_{k4}\\ +x_{k2}{x}_{k4}+x_{k2}{x}_{k5}+x_{k3}{x}_{k5})]=10595.\end{array}$$

$$\begin{array}{l}{W}_{\mathrm{e}}\left(G_4\right)=\sum _{i=1}^{39}{b}_i{d}_i+\sum _{k=1}^2[y_{k1}\left(y_{k2}+y_{k5}\right)+y_{k2}{y}_{k3}+y_{k3}{y}_{k4}+y_{k4}{y}_{k5}+2(y_{k1}{y}_{k3}+y_{k1}{y}_{k4}+y_{k2}{y}_{k4}\\ +y_{k2}{y}_{k5}+y_{k3}{y}_{k5})+e_{k1}\left(e_{k3}+e_{k4}\right)+e_{k2}\left(e_{k4}+e_{k5}\right)+e_{k3}{e}_{k5}+y_{k1}\left(e_{k2}+e_{k4}+2e_{k3}\right)\\ +y_{k2}\left(e_{k3}+e_{k5}+2e_{k4}\right)+y_{k3}\left(e_{k4}+e_{k1}+2e_{k5}\right)+y_{k4}\left(e_{k2}+e_{k5}+2e_{k1}\right)\\ +y_{k5}\left(e_{k3}+e_{k1}+2e_{k2}\right))]=9662.\end{array}$$

$$\begin{array}{l}{\mathit{Sz}}_{\mathrm{v}}\left(G_4\right)=\sum _{i=1}^{39}{e}_i{a}_i{c}_i+\sum _{k=1}^2[e_{k1}\left(x_{k1}+x_{k5}\right)\left(x_{k2}+x_{k3}\right)+e_{k2}\left(x_{k2}+x_{k1}\right)\left(x_{k3}+x_{k4}\right)\\ +e_{k3}\left(x_{k5}+x_{k4}\right)\left(x_{k3}+x_{k2}\right)+e_{k4}\left(x_{k3}+x_{k4}\right)\left(x_{k1}+x_{k5}\right)+e_{k5}\left(x_{k5}+x_{k4}\right)\left(x_{k1}+x_{k2}\right)]=11325.\end{array}$$

$$\begin{array}{l}{\mathit{Sz}}_{\mathrm{e}}\left(G_4\right)=\sum _{i=1}^{39}{e}_i{b}_i{d}_i+\sum _{k=1}^2[e_{k1}\left(y_{k1}+y_{k5}+e_{k4}+e_{k5}\right)\left(y_{k2}+y_{k3}+e_{k2}+e_{k3}\right)+e_{k2}\left(y_{k2}+y_{k1}+e_{k1}+e_{k5}\right)\\ \left(y_{k3}+y_{k4}+e_{k4}+e_{k3}\right)+e_{k3}\left(y_{k3}+y_{k2}+e_{k2}+e_{k1}\right)\left(y_{k4}+y_{k5}+e_{k4}+e_{k5}\right)\\ +e_{k4}\left(y_{k3}+y_{k4}+e_{k2}+e_{k3}\right)\left(y_{k1}+y_{k5}+e_{k1}+e_{k5}\right)\\ +e_{k5}\left(y_{k5}+y_{k4}+e_{k3}+e_{k4}\right)\left(y_{k1}+y_{k2}+e_{k1}+e_{k2}\right)]=10788.\end{array}$$

$$\begin{array}{l}\mathit{PI}\left(G_4\right)=\sum _{i=1}^{39}{e}_i\left(b_i+d_i\right)+\sum _{k=1}^2[e_{k1}\left(y_{k1}+y_{k5}+e_{k4}+e_{k5}+y_{k2}+y_{k3}+e_{k2}+e_{k3}\right)+e_{k2}(y_{k2}+y_{k1}+e_{k1}+e_{k5}\\ +y_{k3}+y_{k4}+e_{k4}+e_{k3})+e_{k3}(\left(y_{k3}+y_{k2}+e_{k2}+e_{k1}+y_{k4}+y_{k5}+e_{k4}+e_{k5}\right)\\ +e_{k4}\left(y_{k3}+y_{k4}+e_{k2}+e_{k3}+y_{k1}+y_{k5}+e_{k1}+e_{k5}\right)+e_{k5}(y_{k5}+y_{k4}+e_{k3}+e_{k4}\\ +y_{k1}+y_{k2}+e_{k1}+e_{k2})]=2552.\end{array}$$

$$\begin{array}{l}\mathit{Mo}\left(G_4\right)=\sum _{i=1}^{39}{e}_i\mid a_i-c_i\mid +\sum _{k=1}^2[e_{k1}\mid \left(x_{k1}+x_{k5}\right)-\left(x_{k2}+x_{k3}\right)\mid +e_{k2}\mid \left(x_{k2}+x_{k1}\right)-\left(x_{k3}+x_{k4}\right)\mid \\ +e_{k3}\mid \left(x_{k3}+x_{k2}\right)-\left(x_{k4}+x_{k5}\right)\mid +e_{k4}\mid \left(x_{k5}+x_{k1}\right)-\left(x_{k4}+x_{k3}\right)\mid \\ +e_{k5}\mid \left(x_{k5}+x_{k4}\right)-\left(x_{k1}+x_{k2}\right)\mid ]=1841\end{array}$$

$$\begin{array}{l}{\mathit{Mo}}_{\mathrm{e}}\left(G_4\right)=\sum _{i=1}^{39}{e}_i\mid b_i-d_i\mid +\sum _{k=1}^2[e_{k1}\mid \left(y_{k1}+y_{k5}+e_{k4}+e_{k5}\right)-\left(y_{k2}+y_{k3}+e_{k2}+e_{k3}\right)\mid \\ +e_{k2}\mid \left(y_{k2}+y_{k1}+e_{k1}+e_{k5}\right)-\left(y_{k3}+y_{k4}+e_{k4}+e_{k3}\right)\mid +e_{k3}\mid \left(y_{k3}+y_{k2}+e_{k2}+e_{k1}\right)\\ -\left(y_{k4}+y_{k5}+e_{k4}+e_{k5}\right)\mid +e_{k4}\mid \left(y_{k3}+y_{k4}+e_{k2}+e_{k3}\right)-\left(y_{k1}+y_{k5}+e_{k1}+e_{k5}\right)\mid \\ +e_{k5}\mid \left(y_{k5}+y_{k4}+e_{k3}+e_{k4}\right)-\left(y_{k1}+y_{k2}+e_{k1}+e_{k2}\right)\mid ]=1982.\end{array}$$

Theorem 6

Let G ₅ be a remdesivir compound. Then, W(G ₅) = 6049, W _e(G ₅) = 5620, Sz _v(G ₅) = 7362, Sz _e(G ₅) = 7067, PI(G ₅) = 1812, Mo(G ₅) = 1221 and Mo _e(G ₅) = 1329.

As we see that remdesivir contains 41 vertices and 44 edges with pentagons and hexagons, it does not admit a Θ‐partition. Let {F _i  : 1 ≤ i ≤ 28} ∪ {S ₁, S ₂} be a Θ^* ‐partition of G ₅ as depicted in Figure 10. The strength‐weighted values of the quotient graph G ₅/F _i are given in Table 6 and e _i = 1; 1 ≤ i ≤ 23 and e _i = 2; 24 ≤ i ≤ 28. The strength‐weighted values of the quotient graph G ₅/S _k , k = 1, 2, are (x ₁₁, x ₁₂, x ₁₃, x ₁₄, x ₁₅) = (24,1,12,2,2); (x ₂₁, x ₂₂, x ₂₃, x ₂₄, x ₂₅) = (3,4,1,1,32); (y ₁₁, y ₁₂, y ₁₃, y ₁₄, y ₁₅) = (24,0,13,1,1); (y ₂₁, y ₂₂, y ₂₃, y ₂₄, y ₂₅) = (2,3,0,0,33); (e ₁₁, e ₁₂, e ₁₃, e ₁₄, e ₁₅) = (1,1,1,1,1); (e ₂₁, e ₂₂, e ₂₃, e ₂₄, e ₂₅) = (2,1,1,1,1). By applying the similar computational procedure of Theorem 5, we complete the proof.

Theorem 7

Let G ₆ be a theaflavin compound. Then, W(G ₆) = 5209, W _e(G ₆) = 5099, Sz _v(G ₆) = 9448, Sz _e(G ₆) = 9660, PI(G ₆) = 1998, Mo(G ₆) = 1210 and Mo _e(G ₆) = 1394.

The number of vertices and edges in theaflavin are 41 and 46, respectively. Since theaflavin contains a cycle on seven vertices, it does not belong to the family of partial cubes. Let {F _i  : 1 ≤ i ≤ 24} ∪ S be a Θ^* ‐partition of G ₆ as shown in Figure 11. The strength‐weighted values of the quotient graph G ₆/F _i are given in Table 7 and e _i = 1; 1 ≤ i ≤ 12, e _i = 2; 13 ≤ i ≤ 22 and e _i = 3; 23 ≤ i ≤ 24. The vertex strength‐weighted values of the quotient graph of G ₆/S are (x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇) = (1,2,2,5,16,1,14); (y ₁, y ₂, y ₃, y ₄, y ₅, y ₆, y ₇) = (0,1,1,4,17,0,15) and edge strength value is depicted in Figure 11C for all the edges.

TABLE 4.

Strength‐weighted values of quotient graph G ₃/F _i

F i	1 ≤ i ≤ 8	9	10	11	12	13	14	15	16	17	18	19	20	21	22
a i	1	8	9	10	12	13	21	7	6	23	22	14	13	11	3
b i	0	8	9	10	12	13	22	7	6	24	23	14	13	11	2
F i	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37
a i	7	7	6	3	4	3	3	3	3	3	3	3	3	4	4
b i	7	7	6	2	3	2	2	2	2	2	2	2	2	3	3

FIGURE 9.

(A) Θ^* ‐classes of ritnovir G ₄ , (B) G ₄/F _i , (C) G ₄/S _k

TABLE 5.

Strength‐weighted values of quotient graph G ₄/F _i

F i	1 ≤ i ≤ 11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
a i	1	2	3	4	6	7	10	9	7	6	5	18	20	21	7
b i	0	1	2	3	5	6	10	9	7	6	5	18	20	21	7
F i	26	27	28	29	30	31	32	33	34	35	36	37	38	39
a i	21	20	18	14	13	11	9	8	3	3	6	3	3	3
b i	21	20	18	14	13	11	9	8	2	2	6	2	2	2

FIGURE 10.

(A) Θ^* ‐classes of remdesivir G ₅ , (B) G ₅/F _i , (C) G ₅/S _k

TABLE 6.

Strength‐weighted values of quotient graph G ₅/F _i

F i	1 ≤ i ≤ 9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28
a i	1	6	7	21	12	19	18	10	11	9	7	6	5	2	2	3	3	3	4	3
b i	0	6	7	21	11	21	20	11	10	8	6	5	4	1	1	2	2	2	3	2

FIGURE 11.

(A) Θ^* ‐classes of theaflavin G ₆ , (B) G ₆/F _i , (C) G ₆/S

TABLE 7.

Strength‐weighted values of quotient graph G ₆/F _i

F i	1 ≤ i ≤ 10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
a i	1	13	13	4	5	9	9	5	4	9	9	17	5	7	7
b i	0	14	14	3	4	9	9	4	3	9	9	18	4	6	6

Now, we compute the topological indices using the cut method as follows.

$$\begin{array}{l}W\left(G_6\right)=\sum _{i=1}^{24}{a}_i{c}_i+x_1\left(x_2+x_7\right)+x_2{x}_3+x_3{x}_4+x_4{x}_5+x_5{x}_6+x_6{x}_7+2(x_1\left(x_3+x_6\right)+x_2\left(x_7+x_4\right)\\ +x_3{x}_5+x_4{x}_6+x_5{x}_7)+3\left(x_1\left(x_4+x_5\right)+x_2\left(x_5+x_6\right)+x_3\left(x_7+x_6\right)+x_4{x}_7\right)=5209.\end{array}$$

$$\begin{array}{l}{W}_{\mathrm{e}}\left(G_6\right)=\sum _{i=1}^{24}{b}_i{d}_i+y_1\left(y_2+y_7\right)+y_2{y}_3+y_3{y}_4+y_4{y}_5+y_5{y}_6+y_6{y}_7+2(y_1\left(y_3+y_6\right)+y_2\left(y_7+y_4\right)\\ +y_3{y}_5+y_4{y}_6+y_5{y}_7)+3\left(y_1\left(y_4+y_5\right)+y_2\left(y_5+y_6\right)+y_3\left(y_7+y_6\right)+y_4{y}_7\right)+27+2(y_1+y_2\\ +y_4+y_5+y_7)+4\left(y_1+y_3+y_4+y_5+y_6\right)+3\left(y_3+y_6\right)+6\left(y_1+y_2+y_7\right)+3(y_2+y_3+y_4\\ +y_5+y_6+y_7)=5099.\end{array}$$

$$\begin{array}{l}{\mathit{Sz}}_{\mathrm{v}}\left(G_6\right)=\sum _{i=1}^{24}{e}_i{a}_i{c}_i+\left(x_7+x_6+x_5\right)\left(x_1+x_2+x_3\right)+\left(x_1+x_7+x_6\right)\left(x_2+x_3+x_4\right)+\left(x_2+x_1+x_7\right)\\ \left(x_3+x_4+x_5\right)+\left(x_3+x_2+x_1\right)\left(x_4+x_5+x_6\right)+2\left(\left(x_4+x_3+x_2\right)\left(x_5+x_6+x_7\right)\right)\\ +\left(x_5+x_4+x_3\right)\left(x_6+x_7+x_1\right)+\left(x_6+x_5+x_4\right)\left(x_7+x_1+x_2\right)=9448.\end{array}$$

$$\begin{array}{l}{\mathit{Sz}}_{\mathrm{e}}\left(G_6\right)=\sum _{i=1}^{24}{e}_i{b}_i{d}_i+\left(y_7+y_6+y_5+4\right)\left(y_1+y_2+y_3+3\right)+\left(y_1+y_7+y_6+3\right)\left(y_2+y_3+y_4+4\right)\\ +\left(y_2+y_1+y_7+3\right)\left(y_3+y_4+y_5+4\right)+\left(y_3+y_2+y_1+3\right)\left(y_4+y_5+y_6+4\right)\\ +2\left(\left(y_4+y_3+y_2+3\right)\left(y_5+y_6+y_7+3\right)\right)+\left(y_5+y_4+y_3+4\right)\left(y_6+y_7+y_1+3\right)\\ +\left(y_6+y_5+y_4+4\right)\left(y_7+y_1+y_2+3\right)=9660.\end{array}$$

$$\begin{array}{l}\mathit{PI}\left(G_6\right)=\sum _{i=1}^{24}{e}_i\left(b_i+d_i\right)+\left(y_7+y_6+y_5+4\right)+\left(y_1+y_2+y_3+3\right)+\left(y_1+y_7+y_6+3\right)\\ +\left(y_2+y_3+y_4+4\right)+\left(y_2+y_1+y_7+3\right)+\left(y_3+y_4+y_5+4\right)+\left(y_3+y_2+y_1+3\right)\\ +\left(y_4+y_5+y_6+4\right)+2\left(\left(y_4+y_3+y_2+3\right)+\left(y_5+y_6+y_7+3\right)\right)\\ +\left(y_5+y_4+y_3+4\right)+\left(y_6+y_7+y_1+3\right)+\left(y_6+y_5+y_4+4\right)+\left(y_7+y_1+y_2+3\right)=1998.\end{array}$$

$$\begin{array}{l}\mathit{Mo}\left(G_6\right)=\sum _{i=1}^{24}{e}_i\mid a_i-c_i\mid +\mid \left(x_7+x_6+x_5\right)-\left(x_1+x_2+x_3\right)\mid +\mid \left(x_1+x_7+x_6\right)-\left(x_2+x_3+x_4\right)\mid \\ +\mid \left(x_2+x_1+x_7\right)-\left(x_3+x_4+x_5\right)\mid +\mid \left(x_3+x_2+x_1\right)-\left(x_4+x_5+x_6\right)\mid \\ +2\mid \left(x_4+x_3+x_2\right)-\left(x_5+x_6+x_7\right)\mid +\mid \left(x_5+x_4+x_3\right)-\left(x_6+x_7+x_1\right)\mid \\ +\mid \left(x_6+x_5+x_4\right)-\left(x_7+x_1+x_2\right)\mid =1210.\end{array}$$

$$\begin{array}{l}{\mathit{Mo}}_{\mathrm{e}}\left(G_6\right)=\sum _{i=1}^{24}{e}_i\mid b_i-d_i\mid +\mid \left(y_7+y_6+y_5+4\right)-\left(y_1+y_2+y_3+3\right)\mid +\mid \left(y_1+y_7+y_6+3\right)\\ -\left(y_2+y_3+y_4+4\right)\mid +\mid \left(y_2+y_1+y_7+3\right)-\left(y_3+y_4+y_5+4\right)\mid +\mid \left(y_3+y_2+y_1+3\right)\\ -\left(y_4+y_5+y_6+4\right)\mid +2\mid \left(y_4+y_3+y_2+3\right)-\left(y_5+y_6+y_7+3\right)\mid +\mid \left(y_5+y_4+y_3+4\right)\\ -\left(y_6+y_7+y_1+3\right)\mid +\mid \left(y_6+y_5+y_4+4\right)-\left(y_7+y_1+y_2+3\right)\mid =1394.\end{array}$$

Theorem 8

Let G ₇ be a nafamostat compound. Then, W(G ₇) = 1951, W _e(G ₇) = 1720, Sz _v(G ₇) = 3198, Sz _e(G ₇) = 2862, PI(G ₇) = 736, Mo(G ₇) = 368 and Mo _e(G ₇) = 418.

We have ∣V(G ₇) ∣  = 26 and ∣E(G ₇) ∣  = 28. Let {F _i  : 1 ≤ i ≤ 19} be the various convex cuts of G ₇ as depicted in Figure 12. The strength‐weighted values of the quotient graph G ₇/F _i are given in Table 8 and e _i = 1; 1 ≤ i ≤ 11, e _i = 2; 12 ≤ i ≤ 18 and e ₁₉ = 3. By simple mathematical calculation based on the cut method, we obtain the required results.

Theorem 9

Let G ₈ be a camostat compound. Then, W(G ₈) = 2846, W _e(G ₈) = 2386, Sz _v(G ₈) = 3856, Sz _e(G ₈) = 3278, PI(G ₈) = 858, Mo(G ₈) = 468 and Mo _e(G ₈) = 510.

Let {F _i  : 1 ≤ i ≤ 24} be a Θ‐partition of G ₈ as shown in Figure 13. The strength‐weighted values of the quotient graph G ₈/F _i are given in Table 9 and e _i = 1; 1 ≤ i ≤ 18 and e _i = 2; 19 ≤ i ≤ 24. Since the number of vertices and edges in camostat are 29 and 30 respectively, we can complete the proof by routine calculations.

Theorem 10

Let G ₉ be umifenovir compound. Then, W(G ₉) = 2100, W _e(G ₉) = 1715, Sz _v(G ₉) = 2878, Sz _e(G ₉) = 2491, PI(G ₉) = 884, Mo(G ₉) = 575 and Mo _e(G ₉) = 634.

We can see that ∣V(G ₉) ∣  = 29 and ∣E(G ₉) ∣  = 31. Let {F _i  : 1 ≤ i ≤ 20} ∪ S be a Θ^* ‐partition of G ₉ as given in Figure 14. The strength‐weighted values of the quotient graph G ₉/F _i are given in Table 10 and e _i = 1; 1 ≤ i ≤ 15, e _i = 2; 16 ≤ i ≤ 20. The strength‐weighted values of G ₉/S are (x ₁, x ₂, x ₃, x ₄, x ₅) = (8,6,9,2,4); (y ₁, y ₂, y ₃, y ₄, y ₅) = (7,5,9,1,3); (e ₁, e ₂, e ₃, e ₄, e ₅) = (1,1,1,1,2). By applying the similar computational procedure of Theorem 5, we obtain the required results.

Theorem 11

Let G ₁₀ be a bevacizumab compound. Then, W(G ₁₀) = 2325, W _e(G ₁₀) = 2242, Sz _v(G ₁₀) = 3574, Sz _e(G ₁₀) = 3534, PI(G ₁₀) = 887, Mo(G ₁₀) = 443 and Mo _e(G ₁₀) = 515.

The number of vertices and edges in bevacizumab are 28 and 31 respectively. Let {F _i  : 1 ≤ i ≤ 17} ∪ S be a Θ^* ‐partition of G ₁₀ as depicted in Figure 14. The strength‐weighted values of the quotient graph G ₁₀/F _i are given in Table 10 and e _i = 1; 1 ≤ i ≤ 9, e _i = 2; 10 ≤ i ≤ 17, c _i =  ∣ V(G ₁₀) ∣  − a _i and d _i =  ∣ E(G ₁₀) ∣  − b _i  − e _i . The strength‐weighted values of G ₁₀/S are (x ₁, x ₂, x ₃, x ₄, x ₅) = (14,3,9,1,1); (y ₁, y ₂, y ₃, y ₄, y ₅) = (14,2,9,0,0); (e ₁, e ₂, e ₃, e ₄, e ₅) = (2,1,1,1,1). We can complete the proof by routine mathematical calculations (Figure 15, Table 11).

FIGURE 12.

(A) Various cuts of nafamostat G ₇ , (B) quotient graph G ₇/F _i

TABLE 8.

Strength‐weighted values of quotient graph G ₇/F _i

F i	1 ≤ i ≤ 5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
a i	1	3	4	3	10	12	13	6	6	10	10	7	7	7	8
b i	0	2	3	2	10	12	13	5	5	10	10	6	6	6	7

FIGURE 13.

(A) Θ‐classes of camostat G ₈ , (B) quotient graph G ₈/F _i

TABLE 9.

Strength‐weighted values of quotient graph G ₈/F _i

F i	1 ≤ i ≤ 7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
a i	1	3	4	10	12	13	10	9	7	6	5	3	7	7	3	13	13	13
b i	0	2	3	10	12	13	9	8	6	5	4	2	6	6	2	12	12	12

FIGURE 14.

(A) Θ^* ‐classes of Umifenovir G ₉ , (B) G ₉/F _i , (C) G ₉/S

TABLE 10.

Strength‐weighted values of quotient graph G ₉/F _i

F i	1 ≤ i ≤ 7	8	9	10	11	12	13	14	15	16	17	18	19	20
a i	1	3	4	5	3	2	6	7	8	9	5	3	3	3
b i	0	2	3	4	2	1	6	7	8	8	4	2	2	2

FIGURE 15.

(A) Θ^* ‐classes of bevacizumab G ₁₀ , (B) G ₁₀/F _i , (C) G ₁₀/S

TABLE 11.

Strength‐weighted values of quotient graph G ₁₀/F _i

F i	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
a i	1	1	2	4	10	11	6	7	8	3	7	3	3	3	3	14	14
b i	0	0	1	3	10	11	6	7	8	2	6	2	2	2	2	14	14

4. WIENER POLARITY INDICES

The Zagreb indices ( M ₁ and M ₂ ) are vertex degree based graph invariants that have been presented during the 1970s and widely concentrated from that point onward. Over the most recent couple of years, an assortment of changes of M ₁ and M ₂ was advanced. It pulled in much intrigue and a plenty of their scientific properties and substance applications were accounted for. The first and second Zagreb [27] indices of a graph G are defined respectively as

$$\begin{array}{l}{M}_1\left(G\right)=\sum _{\mathit{uv}\in E\left(G\right)}\left[d_u+d_v\right]=\sum _{v\in V\left(G\right)}{d}_v^2,\\ M_2\left(G\right)=\sum _{\mathit{uv}\in E\left(G\right)}\left[d_u\times d_v\right].\end{array}$$

The Wiener polarity index of a graph G is defined as

$$W_{\mathrm{P}}\left(G\right)=\mid \left\{\left\{u,v\right\}:d_G\left(u,v\right)=3,u,v\in V\left(G\right)\right\}\mid .$$

It was observed [28] that $W_{\mathrm{P}}\left(G\right)=\frac{1}{2}\sum _{v\in V\left(G\right)}\mid N_G^3\left(v\right)\mid$ and the importance of W _P has been demonstrated in various papers (see References [29, 30] for details). Suppose G is an acyclic graph, it was realized [31] that

$$W_{\mathrm{P}}\left(G\right)=\sum _{\mathit{uv}\in E\left(G\right)}\left(d_u-1\right)\times \left(d_v-1\right).$$

If G is a C ₃ ‐free and C ₄ ‐free graph such that its different cycles have at most one common edge [30],

$$W_{\mathrm{P}}\left(G\right)=M_2\left(G\right)-M_1\left(G\right)+\mid E\left(G\right)\mid -5N_{\mathrm{P}}\left(G\right)-3N_{\mathrm{h}}\left(G\right),$$

and that if G is a C ₃ ‐free graph such that its different cycles have at most one common edge [29],

$$W_{\mathrm{P}}\left(G\right)=M_2\left(G\right)-M_1\left(G\right)+\mid E\left(G\right)\mid -5N_{\mathrm{p}}\left(G\right)-3N_{\mathrm{h}}\left(G\right)-\sum _i\left(K_i-4\right)N_{q_i}\left(G\right),$$

where $N_{q_i}\left(G\right)$ is the number of quadrangles of type i such that the sum of degrees on the vertices of that type quadrangle as K _i . In this section, we compute the Wiener polarity indices of antiviral drug compounds used for the treatment of COVID‐19.

Theorem 12

Let G _i , 1 ≤ i ≤ 10 be the various chemical compounds used for COVID‐19. Then, W _P(G ₁) = 31, W _P(G ₂) = 32, W _P(G ₃) = 70, W _P(G ₄) = 68, W _P(G ₅) = 69, W _P(G ₆) = 85 W _P(G ₇) = 39, W _P(G ₈) = 39, W _P(G ₉) = 49 and W _P(G ₁₀) = 42.

Since ∣E(G ₁) ∣  = 30, N _p(G ₁) = 0 and N _h(G ₁) = 2, we have

Similarly we can obtain the other values by using Table 12.

TABLE 12.

Edge partition, Zagreb indices, number of pentagons and hexagons

Graph G	Edge partition of G i								M 1(G)	M 2(G)	N p	N h
	(1, 3)	(2, 2)	(2, 3)	(3, 3)	(1, 4)	(1, 2)	(3, 4)	(2, 4)
G 1 [5]	2	5	12	2	‐	2	‐	‐	106	120	0	2
G 2 [5]	2	6	12	2	‐	2	‐	‐	110	124	0	2
G 3	8	14	20	7	‐	‐	‐	‐	230	263	0	4
G 4	10	11	25	5	‐	1	‐	‐	242	271	2	1
G 5 [5]	5	9	14	6	2	2	2	4	216	257	2	2
G 6 [5]	10	‐	22	14	‐	‐	‐	‐	234	288	0	5
G 7	5	4	16	3	‐	‐	‐	‐	134	154	0	3
G 8	7	5	16	2	‐	‐	‐	‐	140	155	0	2
G 9	6	6	9	9	‐	1	‐	‐	150	179	1	2
G 10	1	10	15	4	‐	1	‐	‐	146	171	1	3

5. CONCLUSION

At present, the utilization of topological descriptors covers a large portion of the principle research regions of drug development, for example, lead discovery and lead optimization. At the point when topological descriptors are joined with another quantum chemically determined electronic parameters, for example, highest occupied molecular orbital lowest unoccupied molecular orbital energy gaps, hardness, polarizability, atomic electrostatic potentials, natural bond orbital investigation, and so on, one could get quantitative proportions of the relative stabilities, reactivities, and binding potentials of a drug. In this paper, we have derived the quantitative structural properties of several antiviral drug compounds used to fight the outbreak of COVID‐19. Our computation method is based on the identification of suitable edge cuts such that the resulting graph leaves many convex components and then derive the required properties. A graphical representation of computed topological indices such as Wiener, Szeged, Mostar and their variants is depicted in Figure 16. Finally, we believe that our results can be useful in finding the exact medication for COVID‐19.

FIGURE 16.

A graphical representation of topological indices of COVID‐19 drug compounds

AUTHOR CONTRIBUTIONS

Jia‐bao Liu: Conceptualization; methodology; supervision. Micheal Arockiaraj: Conceptualization; supervision; validation. Arulperumjothi M: Software; validation; writing‐original draft. Savari Prabhu: Investigation; visualization; writing‐review and editing.

Liu J‐B, Arockiaraj M, Arulperumjothi M, Prabhu S. Distance based and bond additive topological indices of certain repurposed antiviral drug compounds tested for treating COVID‐19. Int J Quantum Chem. 2021;121:e26617. 10.1002/qua.26617

DATA AVAILABILITY STATEMENT

All data required for this research work are within the manuscript.