Web of Science (Emerging Sources Citation Index), Scopus, ISC

Document Type : Original Research Article


1 Department of Mathematics, Bangalore University, Jnana Bharathi Campus, Bangalore-560 056, India

2 Department of Mathematics, Gulbarga University, Gulbarga 585106, India


In this paper, we used linear, quadratic, cubic, logarithmic, and exponential regression models to analyze the statistical properties of certain important molecular structures such as chloroquine, hydroxychloroquine, and remdesivir, taking into account various forms of augmented Zagreb indices. The research could be a fresh attempt to improve QSPR model prediction analysis using the above molecular descriptors, which are used to research chemical, medical, and pharmacological qualities.

Graphical Abstract

Different types of augmented Zagreb indices of some chemical drugs: A QSPR model


Main Subjects


Given the rapid technological advancement, many medical and pharmaceutical solutions are being developed consistently, which requires a significant amount of work to examine the biological and physiochemical properties of these drugs. Chloroquine, hydroxychloroquine, and remdesivir are all effective treatments for COVID-19 patients. Scientists have discovered that both chloroquine and remdesivir can prevent the virus from multiplying in cells during the early stages of a life-threatening situation [5, 6,9,11,24,26,27,28,29]. The main motivation for the work is to increase the efficiency of the drugs.

Let G = (V, E) be a simple, finite and connected graph. The degree d(u) of a vertex u is the number of vertices adjacent to v. The edge connecting the vertices u and v will be denoted by uv.  We refer to [19] for undefined term and notation.

In the modelling of mathematics, a molecular graph or a chemical graph is a simple graph related to the structure of a chemical compound. Each vertex of this graph represents an atom of the molecule, and its edges represent the bonds between atoms. A topological index is a numerical parameter mathematically derived from the graph structure. These topological indices are useful for establishing correlations between the structure of a molecular compound and its physicochemical properties [17].

In [10], Furtula et al. introduced the augmented Zagreb index of a graph, which is defined as

This topological index has been found to be a useful predictive indicator in the research on heat generation in octanes and heptanes, with a prediction power that is superior to the atom bond connectivity index [10]. This index has also been researched in the past, such as in [20,21].

In [22], the second, third and fourth augmented Zagreb indices of G were introduced by Kulli as follows:

where the edge uv of a graph G has a number nu of vertices that are closer to the vertex u than to the vertex v.

For a molecular graph G, the third augmented Zagreb index is defined as:

where the number mu of edges of G lies closer to the vertex u than to the vertex v for the edge uv of G.

The fourth augmented Zagreb index of a molecular graph G is defined as:

where the number e(u) is the eccentricity of all vertices adjacent a vertex u.

In [15], the Sanskruti index (or fifth augmented Zagreb index) of G was introduced, which is defined as: 

where the number s(u) is the sum of the degrees of all vertices adjacent to a vertex u.

Some novel versions of topological indices have recently been investigated [1,2,3, 4,7,13,17]. For chemical structures, see [8,23].

In this paper, we are trying to analyze these three chemical drugs by using augmented Zagreb types of indices by different types of regression models.


Andersag (1934) discovered chloroquine as an antiviral chemical (drug). This medicine is primarily used to treat and prevent malaria. Let G1 be the molecular structure of chloroquine with 21 atoms and 23 bonds as shown in Figure 1.

In the following theorem, we compute the several forms of augmented Zagreb indices of chloroquine.

Theorem 1. Let G1 stand for chloroquine's molecular structure. Then

(i)  AZI(G1) = 162.66935484

(ii) AZI2(G1) = 1275.4119738

(iii) AZI3(G1) =1270.8788547

(iv) AZI4(G1) = 4132.9250581

(v) AZI5(G1) = 802.23818748

Proof: Using G1's bond partition definitions and cardinalities, we deduce


 Scientists developed hydroxychloroquine, a less toxic form of chloroquine, in 1946, and it was later used to treat a variety of disorders. Let G2 be the molecular structure of hydroxychloroquine with 22 atoms and 24 bonds as shown in Figure 2.

In the following theorem, we compute the several forms of augmented Zagreb indices of hydroxychloroquine.

Theorem 2. Let G2 stand for hydroxychloroquine's molecular structure.

(i)AZI(G2) = 170.546875

  1. ii) AZI2(G2) = 1372.7191427

(iii) AZI3(G2) = 1691.8960138

(iv) AZI4(G2) = 4821


(v) AZI5 (G2) = 883.90022452

Proof: Using G2's bond partition definitions and cardinalities, we deduce


The first drug to get emergency approval from the food and drug administration under COVID-19 is Remdesivir [14], which is an intravenous nucleotide prodrug of an adenosine analog. Remdesivir works by blocking the RNA polymerase. Let G3 be the molecular structure of Remdesivir with 41 atoms and 44 bonds as shown in Figure 3.

In the following theorem, we compute the several forms of augmented Zagreb indices of Remdesivir.

Theorem 3. Let G3 stand for Remdesivir molecular  structure. Then

(i) AZI(G3) = 349.60749074

(ii) AZI2(G3) = 8023.0266267

(iii) AZI3(G3) = 11325.870309

(iv) AZI4(G3) = 19860.159994

(v) AZI5 (G3) = 1996.9680634

Proof: Using G3's bond partition definitions and cardinalities, we deduce

Data set and computed values

In order to find the usefulness of a topological index, we have to predict regression models between the physicochemical properties and the calculated topological indices. In Tables 4 and 5, we have tabulated the calculations of the above topological indices and the physicochemical properties of molecular structures, respectively. These va)lues are useful for creating regression models. The data set of the above-mentioned molecular structures consists of the following physicochemical properties as given in Table 5 found at ChemSpider.

Regression models

Regression models are used to fit the curves. Accordingly, we studied linear, quadratic, cubic, logarithmic, and exponential regression models. We constructed regression models of the above-mentioned topological indices with the physicochemical properties of molecular structures as shown in Table 5. In the regression model table, we considered the square of the coefficient of the correlation (R2), the F-ratio test, and significance (sig). The model with the maximum R2 is the best predictor or goodness of fit of the regression model. For the model to be efficient, if the F-ratio test is greater than one and the sig value is less than 0.05, then the topological indices reliably predict the dependent variable for the particular physicochemical property. The regression models are obtained from Tables 4 and 5 with SPSS statistical software, as shown in Tables 6, 7, and 8.

Here, we have shown a few best predictors of the topological index regression models for the particular physicochemical property.


The QSPR study has shown that molecular descriptors (topological indices) are the best tools to predict the physicochemical properties of drugs used for chemical, medical, and pharmaceutical characteristics. In the linear regression model, all molecular descriptors are best predicted with the mentioned physicochemical properties, except surface tension and logp. In a quadratic regression model, molecular descriptor AZI2 is best predicted with polarizability. In a logarithmic regression model, molecular descriptor AZI5 is best predicted with polar surface area. In an exponential regression model, molecular descriptors AZI and AZI3 are best predicted with molar refractivity. The results of the above study may be used in the further development of drugs used for chemical, medical, and pharmaceutical characteristics.



Authors’ contributions

All of the authors have made major contributions to this paper, and they have all given their approval to the final version. The final manuscript was read and approved by all the writers.

Conflict of interest

No competing interests have been declared by the authors.


T.V. Asha: https://orcid.org/0000-0003-2275-3532

V.R. Kulli: https://orcid.org/0000-0002-6881-5201

B. Chaluvaraju: https://orcid.org/0000-0002-4697-0059


How to cite this article: T.V. Asha, V.R. Kulli, B. Chaluvaraju*. Different types of augmented Zagreb indices of some chemical drugs: A QSPR model. Eurasian Chemical Communications, 2022, 4(6), 513-524. Link: http://www.echemcom.com/article_147154.html


Copyright © 2022 by SPC (Sami Publishing Company) + is an open access article distributed under the Creative Commons Attribution License(CC BY)  license  (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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