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Document Type : Original Research Article


1 Department of Mathematics, K.S. School of Engineering and Management (Affiliated to ‎Visvesvaraya Technological University), Bengaluru-560109, India‎

2 Department of Mathematics, Adichunchanagiri Institute of Technology (Affiliated to ‎Visvesvaraya Technological University), Chikamagalur-577102, India‎

3 Department of Mathematics, Malnad College of Engineering (Affiliated to Visvesvaraya Technological University), Hassan-573201, India


In the study of graph properties, topological indices and graph labelling are both enormous topics. In this paper, we connect the ideas of topological indices with graph labelling, resulting in a number of novel topological indices to study the labelled graphs. We present new topological indices for certain molecular graphs that admit cordial labelling in this article. Through topological indices, graph theory is playing an essential part in QSPR data analysis. In this paper, we consider the labelled square index SQI(G), labelled product Index PI(G), labelled sum Index SI(G), labelled Nirmala Index NI(G ), labelled Sombor Index SOLI(G), labelled forgotten Index FI(G) and Cluster of all these Indices.

Graphical Abstract

Cordial labelling of molecular structures and labelled topological indices of molecular graphs; a qspr model


Main Subjects


Chemical graph theory is a branch of mathematics concerned with chemistry that blends mathematical design and graph theory to study chemical processes. It focuses on topological indices which have been closely connected with chemical molecules and molecular characteristics. Topological indices are frequently utilized in the structure-activity relationship/ quantitative structure-property (QSAR/ QSPR) design to predict the characteristics of a molecule or molecules.

Let G(V,E) be a simple connected graph with V(G) as the vertex set and E as the edge set E(G). A molecular graph is a figure that used to represent synthesized good in addition to express the drug's chemical structure. We refer to [1] for any additional concepts or terms.

A molecular descriptor emphasizes in providing the most accurate numerical representation of potential molecule form. The most commonly used molecular descriptors are molecular connectivity indices. These molecular meters are often referred to it as topological indices because it describes the topology of a molecule. They are considered like graph invariants since their concepts are based on notions from graph theory. In theoretical chemistry, their specific features have been studied and have found with considerable form particularly in QSPR/QSAR/QSTR research [2-12].

A molecular graph depicts the unsaturated hydrocarbon skeletons of molecules and their compounds. Its edges indicate covalent links between non-hydrogen atoms, while its vertices represent non-hydrogen atoms. Molecular graphs have important functions in chemoinformatics [13], quantitative structure-property relationships(QSPR), quantitative structure-activity relationships (QSAR), virtual screening of chemical libraries and computational drug design.

Drugs are often considered as a critical tool for preventing and controlling diseases. Drug development is a time-consuming, difficult and costly procedure. In the field of drug development, computer-aided drug design plays an essential role. This involves predicting target candidates’ electronic, drug-like pharmacokinetic, 3D-QSAR and physicochemical characteristics.

Molecular structures of some drugs

Chloroquine is the most widely used medication to treat malaria. It is also used to treat autoimmune illnesses. Chloroquine inhibits DNA replication and RNA transcription by interfering with nuclear proteins .It is an antimalarial drug that is taken orally [14,15], has more information.

Hydroxychloroquine possesses antiviral properties which are quite similar to chloroquine. Both have immune modifying properties that can boost their antiviral efficacy in vivo. These drugs dampen the cytokine storm by blocking T cell activation, which reduces COVID-19's acute evolution. See [15,16] for more information and it was Hydroxychloroquine is made from 4-aminoquinoline. Since World War II, it has been utilized as an antimalarial medication. It is also used to treat skin problems, lupus erythematosus, rheumatoid arthritis, and other inflammatory conditions.

Remdesivir is an antiviral medication that is being tested to prevent Ebola virus infection. In addition, it is a medication that can fight a variety of viruses. Remdesivir is a nucleotide analogue medication which prevents the replication of viral RNA. It is being studied as a COVID-19 treatment and has been approved for emergency use in the United States, India, and Singapore, as well as for patients with severe symptoms in Japan, the European Union, the United Kingdom, and Australia. The clinical trial is now taking place at many hospitals, and efficacy testing is pending. See [17,18,19, and 20] for further information.

Ribavirin is an antiviral drug that treats hepatitis C, RSV, and certain viral hemorrhagic fevers. It is used in the treatment of hepatitis C alongside other drugs like simeprevir, peginterferon alfa-2b, sofosbuvir and peginterferon Alfa 2a. For Ebola or Marburg infections, it should not be used. Ribavirin can be breathed or given by mouth. Ribavirin was first used in 1986 after being invented in 1971. For more details, See [21,22]

Favipiravir is an anti-RNA virus pyrazine carboxamide derivative. Favipiravir works by inhibiting an enzyme called RNA-dependent RNA polymerase, which is involved in viral genome reproduction and replication. Toyama Chemical Co., Ltd created this medicine for the treatment of influenza A and B in Japan and it is only approved for use there. Furthermore, it is being experimented for the therapy of COVID-19 and viral hemorrhagic fever (Ebola). For more details, one can refer to [23,24,25].

Thalidomide is used to treat psoriasis systemic lupus erythematosus and gastrointestinal inflammatory illnesses, among other autoimmune conditions and it causes congenital abnormalities (phocomelia) in the fetus [26].

2-phenoxyethanol is the aromatic ether with a 2-hydroxyethyl group substituting on oxygen. It works as an anti-infective and a depressant for the central nervous system. It’s glycol ether, an aromatic ether, and a primary alcohol. It is mainly composed of phenol.

Chloroquine, Hydroxychloroquine, Remdesivir, Lopinavir, Ritonavir, Arbidol, Theaflavin, and Thalidomide are all possible COVID-19 treatments [27]. To forecast boiling point, molar refractivity, surface tension, polarizability, polar surface area and molar volume of these medicines, the most relevant topological indices and curvilinear regression studies are achieved.

Alsinai et al. [28] introduced Hdr degree based indices and mhr-polynomial for the treatment of covid-19. Ammar Alsinai et al. [29] introduced Reciprocal leap indices of some wheel related graphs. Arkhanda Afzal et al. [30] introduced topological aspects of silicate network using M-polynomial. Alsinai et al. [31] worked on fourth leap Zagreb index of graphs. Hasan et al. [32] introduced distance and Degree based Topological Polynomial and Indices of X-Level Wheel Graph.

  1. Princess Rathinabai et al. [33] introduced and defined the density-based topological indices. Motivated by this paper, we are extending this concept to labelled based topological indices for the above molecular structures.

Definition 1.1 [33]: Labelled Incident of vertex, (u), with regard to a labeling of a labelled graph G, is defined as (u)=   f(uv) is the label allocated to the edge uv. In other words, labelled Incident of vertex u is the sum of all the labels of the edges that intersect with u.

Definition 1.2 [34]: A labelled graph is a graph with labels applied to its vertices and edges based on some particular concept. We construct some distinct types of new topological indices in this part and introduce the concept of a vertex incident in a labelled graph. Only non-negative integers are used as labels in the introduction of new topological indices.

Definition 1.3 [35]. Let f be such that f: V (G) → {0, 1} and for each edge uv assign the label |f(u) − f(v)|. A binary vertex labeling of a graph G is called cordial labeling if and , where  denote the number of vertices and edges of G with label i(= 0 or 1), respectively. If it admits cordial labeling then graph G is cordial.

The followings (Table 1) are the notations used in the subsequent section: For more details follow [36,37,38,39,40,41,42,43,44].



Regression models

To fit the curves, regression models are used. Accordingly, we will look at linear, quadratic, cubic, logarithmic, and exponential regression models in this article. We may see the squared coefficient of correlation ( ), F-ratio test, and significance in the regression model table (sig). The best predictor or goodness of fit of the regression model is the maximum ( ), F-ratio test should be larger than once for efficient model and significance value should be less than 0.05, then topological indices reliably predict the dependent variable for the specific physicochemical feature.

Main results

The linear, quadratic, exponential, and cubic regression models are obtained by using the data in Tables 2 and 3 with the SPSS software. Tables 4, 5, and 6 shows the square of the correlation coefficient  obtained by the linear regression model between various topological indices and physicochemical properties of chloroquine, hydroxychloroquine, a Favipiravir, remdesivir, ribavirin, and thalidomide drugs used in the treatment of COVID-19 patients. In the following, a few best predictors of the topological index regression models were indicated for the particular physicochemical property.

It is evident that Figures 8-10 depict the plots of linear, logarithmic, cubic, quadratic, and exponential regression models of the molar refractivity with SOLI(G),NLI(G),SI(G). Figure 11 demonstrates the regression models of the polar surface with CSI(G).



Figure 12 displays the plots of linear, logarithmic, cubic, quadratic, and exponential regression models of the polarizability with NLI(G). Figure 13 shows the regression models of the molar volume with CSI.

Figure 14 indicates the plots of linear, logarithmic, cubic, quadratic, and exponential regression models of the surface tension with CSQI (G) and Figure 15 demonstrates the regression models of boiling point with CFI (G).


Taken into account numerous variants of labelled topological indices, the regression models were used to analyze the molecular structure in this paper. The work could be a fresh effort to improve QSPR model prediction outcomes utilizing the labelled topological indices, which assist chemists come up with new medication design concepts. The study involves the use of favipiravir, ribavirin, remsedivir, chloroquine, and hydroxychloroquine. The molecular descriptors of those structures were originally discovered. The computed values of those components were then summarized.

According to the QSPR study, molecular descriptors [13] (topological indices) are the most effective instruments for predicting physicochemical qualities of pharmaceuticals employed for chemical, medical and pharmaceutical purposes. The regression models can be correlate to study the relationship between two variables like molecular descriptor and physicochemical properties. The molecular descriptors like NLI(G), CSQI(G), and SI(G) are best predicted with physicochemical properties like molar refractivity and polarizability. The molecular descriptor CSQI(G) is best predicted with surface tension. The findings of the aforementioned study could be applied to the development of new medications with chemical, medicinal, and pharmaceutical properties.

Disclosure statement

The authors declare that there is no conflict of interest regarding the publication of this article.


The authors extend gratitude to the referees for their careful peer review and    helpful comments.

Author’s contributions

All 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 its authors.

Conflict of interest

No competing interests have been declared by the author


Vinutha S.V.: http://orcid.org/0000-0002-5576-6316

Shrikanth A.S.: https://orcid.org/0000-0003-4463-0533


How to cite this article: Vinutha S.V., ‎Shrikanth A.S.*, ‎Nagalakshmi A. Cordial labelling of molecular structures and labelled topological indices of molecular graphs; a qspr model. Eurasian Chemical Communications, 2022, 4(11), 1087-1107. Link: http://www.echemcom.com/article_152699.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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