Degree-based Indices Research Articles

Neighbourhood Sum-Based Structural Analysis for Sodalite System

Background: Sodalite is a type of zeolite with an intricate structure comprising a system of interrelated cages and tunnels. It is extensively used in sieving applications due to its unique structure and properties. As a result, it finds several uses in water and air purifica-tion, radioactive decontamination, detergents, and so on. Due to the potential positive envi-ronmental impact of sodalite materials, analysing the molecule at a structural level becomes the need of the hour. Methods: Molecular descriptors form the basis of many convenient and cost-effective tech-niques for studying molecular structures. In this article, the neighbourhood sum-based de-scriptors are computed edge-partition techniques, simplifying the intricate structures with cages and tunnels into more simple graphs for mathematical convenience. Results: This article presents the calculated analytical expressions for molecular descriptors, specifically focusing on various neighbourhood sum degree-based indices in sodalite struc-tures. Conclusion: The results presented in this article establish the dependence of the physical properties of a molecule on its underlying structure using the computed molecular de-scriptors. The graphical comparison of the results provides a visual representation of the be-haviour of indices with respect to the molecular structure.

Degree-Based Topological Indices of the Benzenoid Circumcoronene Series

This paper deals with some types of topological indices called valency-based indices or degree-Based Indices. Specifically, Multiplicative Forgotten, Multiplicative Yemen, modified Forgotten, modified Yemen, generalized modified first Zagreb, generalized modified sum connectivity, and generalized modified product connectivity indices of the benzenoid circumcoronene series are computed. Moreover, the formulas of polynomials for all these topological indices were derived.

Comparative study of Sombor index and its various versions using regression models for top priority polycyclic aromatic hydrocarbons

The aromatic compounds having structural configurations with two or more fused benzene rings are the polycyclic aromatic hydrocarbons (PAHs). Topological indices are valuable tools for studying the structure property relationships of PAHs and also helps in predicting various properties and activities. They find applications widely in computational chemistry, drug design and QSPR studies. This article focuses on analysing the potential predictive index for Sombor index (SO), elliptic Sombor index (ESO), Euler Sombor index (EU), reverse Sombor index (RSO), reverse elliptic Sombor index (RESO) and reverse Euler Sombor index (REU) using regression models for top priority 38 PAHs. From the study it is evident that, SO and RSO have proved to be potential predictive indices among the considered degree-based and reverse degree-based indices. The variation of best predictive index with minimal RMSE are plotted for linear, quadratic and cubic regression models for better understanding.

QSPR/QSAR study of antiviral drugs modeled as multigraphs by using TI’s and MLR method to treat COVID-19 disease

The ongoing COVID-19 pandemic continues to pose significant challenges worldwide, despite widespread vaccination. Researchers are actively exploring antiviral treatments to assess their efficacy against emerging virus variants. The aim of the study is to employ M-polynomial, neighborhood M-polynomial approach and QSPR/QSAR analysis to evaluate specific antiviral drugs including Lopinavir, Ritonavir, Arbidol, Thalidomide, Chloroquine, Hydroxychloroquine, Theaflavin and Remdesivir. Utilizing degree-based and neighborhood degree sum-based topological indices on molecular multigraphs reveals insights into the physicochemical properties of these drugs, such as polar surface area, polarizability, surface tension, boiling point, enthalpy of vaporization, flash point, molar refraction and molar volume are crucial in predicting their efficacy against viruses. These properties influence the solubility, permeability, and bio availability of the drugs, which in turn affect their ability to interact with viral targets and inhibit viral replication. In QSPR analysis, molecular multigraphs yield notable correlation coefficients exceeding those from simple graphs: molar refraction (MR) (0.9860), polarizability (P) (0.9861), surface tension (ST) (0.6086), molar volume (MV) (0.9353) using degree-based indices, and flash point (FP) (0.9781), surface tension (ST) (0.7841) using neighborhood degree sum-based indices. QSAR models, constructed through multiple linear regressions (MLR) with a backward elimination approach at a significance level of 0.05, exhibit promising predictive capabilities highlighting the significance of the biological activity IC50\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$IC_{50}$$\\end{document} (Half maximal inhibitory concentration). Notably, the alignment of predicted and observed values for Remdesivir’s with obs pIC50=6.01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${pIC_{50} = 6.01}$$\\end{document},pred pIC50=6.01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${pIC_{50} = 6.01}$$\\end{document} (pIC50\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$pIC_{50}$$\\end{document} represents the negative logarithm of IC50\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$IC_{50}$$\\end{document}) underscores the accuracy of multigraph-based QSAR analysis. The primary objective is to showcase the valuable contribution of multigraphs to QSPR and QSAR analyses, offering crucial insights into molecular structures and antiviral properties. The integration of physicochemical applications enhances our understanding of factors influencing antiviral drug efficacy, essential for combating emerging viral strains effectively.

Structure-property modeling of physicochemical properties of fractal trigonal triphenylenoids by means of novel degree-based topological indices.

Trigonal triphenylenoids (TTPs) are a fascinating class of organic molecules with unique structural and electronic properties. Their diverse applications, ranging from organic electronics to nonlinear optics, have spurred significant research interest in understanding their physicochemical behavior. Topological indices, mathematical descriptors derived from the molecular graph, offer valuable insights into the structural complexity and potential properties of TTPs. This work focuses on exploring the utility of degree-based topological indices in characterizing and predicting the properties of trigonal triphenylenoids. We systematically calculate various degree-based topological indices, for a diverse set of TTPs with varying substituents and topologies. The relationships between these indices and key physicochemical properties, such as HOMO-LUMO energy gap, thermodynamic stability, and reactivity are investigated using statistical and machine learning approaches. We identify significant correlations between specific degree-based indices and different properties, allowing for potential prediction of these properties based solely on the topological information.

On QSAR modeling with novel degree-based indices and thermodynamics properties of eye infection therapeutics.

Topological descriptors are numerical results generated from the structure of a chemical graph that are useful in identifying the physicochemical characteristics of a wide range of drugs. The introduction of molecular descriptors advances quantitative structure-property relationship research. This article focuses on the nine degree-based topological indices and the linear regression model of the eye infection drugs. We introduced two new indices, namely, the "first revised Randic index" and the "second revised Randic index, for the analysis of eye infection drugs. Topological indices are calculated by using edge partitioning, vertex degree counting, and vertex degree labeling. This analysis is done with a scientific calculator and then authenticated with Matlab, a potent tool for examining data. The experimental data and results of the topological indices serve as inputs for the statistical computations and provide the values of intercepts, slopes, and correlation coefficients. All the correlations for the eye-infection drugs are positive, indicating a direct relationship between the experimental and estimated results of the drugs. There are significant results of the p-test for all of the characteristics of eye infection, such as molecular weight, boiling point, enthalpy, flash point, molar refraction, and molar volume, that validate the accuracy of the computations. A significant link was determined in this study between the defined indices with two properties: molar weight and molar refraction. The molar weight and molar refraction have a correlation coefficient ranging from 0.9. These results demonstrate a strong association between the indices and the properties under investigation. The linear regression approach is a valuable tool for chemists and pharmacists to obtain data about different medicines quickly and cost-effectively.

Estimating physico-chemical properties of drugs for prostate cancer using degree-based and neighbourhood degree-based topological descriptors

Prostate cancer is one of the most alarming diseases among men and leads to various illnesses. The main flaw in this disease is that there is not a single test that can find it. Prostate-specific antigens and prostate biopsies are used to identify this type of cancer. Many pharmaceutical industries do active research to identify non-steroidal drugs that can be used to treat prostate cancer effectively. QSPR analysis is one of the best practices that aid scientists in getting a deep insight into the physico-chemical properties of drugs without actually performing experiments in a wet lab, thereby cutting down on the cost of manufacturing such drugs. In this article, we perform a QSPR analysis of nineteen drugs primarily used to treat prostate cancer by using a set of ten degree-based indices and another set of ten novel neighbourhood-degree-based indices. In addition, we have done a comparative study of the findings obtained through these two different sets of indices over thirteen physico-chemical properties of prostate cancer drugs.

Structure-property modeling of coumarins and coumarin-related compounds in pharmacotherapy of cancer by employing graphical topological indices.

Coumarins, a subgroup of colorless and crystalline oxygenated heterocyclic compounds originally discovered in the plant Dipteryx odorata, were the subject of a recent study investigating their quantitative structure-activity relationship (QSAR) in cancer pharmacotherapy. This study utilized graph theoretical molecular descriptors, also known as topological indices, as a numerical representation method for the chemical structures embedded in molecular graphs. These descriptors, derived from molecular graphs, play a pivotal role in quantitative structure-property relationship (QSPR) analysis. In this paper, intercorrelation between the Balban index, connective eccentric index, eccentricity connectivity index, harmonic index, hyper Zagreb index, first path Zagreb index, second path Zagreb index, Randic index, sum connectivity index, graph energy and Laplacian energy is studied on the set of molecular graphs of coumarins. It is found that the pairs of degree-based indices are highly intercorrelated. The use of these molecular descriptors in structure-boiling point modeling was analyzed. Finally, the curve-linear regression between considered molecular descriptors with physicochemical properties of coumarins and coumarin-related compounds is obtained.

On topological analysis of two-dimensional covalent organic frameworks via M-polynomial

Covalent organic frameworks (ZnP-COFs) made of zinc-porphyrin have become effective materials with a variety of uses, including gas storage and catalysis. To simulate the structural and electrical features of ZnP-COFs, this study goes into the computation of polynomials utilizing degree-based indices. We gave a methodical study of these polynomial computations using Excel, illustrating the complex interrelationships between the various indices. Degree-based indices provide valuable insights into the connectivity of vertices within a network. M-polynomials, on the other hand, offer a mathematical framework for representing and studying the properties of 2D COFs. By encoding structural information into a polynomial form, M-polynomials facilitate the calculation of various topological indices, including the Wiener index, Zagreb indices, and more. The different behavior of ZnP-COFs based on degree-based indices was illustrated graphically, and this comparison provided insightful information for prospective applications and the construction of innovative ZnP-COF structures. Moreover, we discuss the relevance of these techniques in the broader context of materials science and the design of functional covalent organic frameworks.

Degree-based indices of hypergraphs: Definitions and first results

For a [Formula: see text]-uniform hypergraph [Formula: see text], we introduce degree-based indices such as the general sum-connectivity index [Formula: see text] and the general Randić index [Formula: see text], where [Formula: see text], [Formula: see text] is the set of hyperedges of [Formula: see text] and [Formula: see text] is the degree of a vertex [Formula: see text] in [Formula: see text]; [Formula: see text] and [Formula: see text]. Other indices such as the first and second Zagreb index, first and second hyper-Zagreb index, classical sum-connectivity index, classical Randić index and harmonic index of a hypergraph [Formula: see text] are special cases of the general indices. For [Formula: see text], we obtain upper bounds on [Formula: see text] and [Formula: see text] for a uniform hypergraph [Formula: see text] with given order, order and number of isolated vertices, order and maximum degree, order and diameter at least [Formula: see text], and lower bounds for uniform hypergraphs with given order and no isolated vertices, order and minimum degree, and order and maximum possible degree. We also present extremal graphs for all the bounds. Bounds on Zagreb indices follow from our results on general indices.

Atom-bond-connectivity (ABC) indices of graphene sheets, zigzag single walled carbon nanotubes and single walled carbon nanotori

Abstract Atom-bond-connectivity (ABC) indices are obtained in analytical forms for graphene sheets, zigzag single walled carbon nanotubes (SWCNTs), and single walled carbon nanotori in terms of number of rings (r) that measures the length and the number of hexagons in between two rings (h) that dictates the width of the concerned systems. The procedures followed for ABC index have been used to obtain the expressions of augmented Zagreb and Randić indices for such systems. Logarithm of ABC indices of zigzag SWCNTs are found to correlate linearly well with the bond dissociation energies per C–C bond and the Young’s moduli of said SWCNTs with fixed number of rings (r) but varying number of hexagons (h) in between two successive rings. The plot of logarithm of ABC index versus Young’s modulus of such SWCNTs in varying both r and h simultaneously is not a straight line but fits well with the sigmoidal (Boltzmann) curve. Wiener index, one of the important distance based index, has recently been found to have similar correlations with the concerned properties of such systems. Similar plots would appear for the said properties of the zigzag SWCNTs with other degree-based indices like augmented Zagreb and Randić indices, as have been indicated from their respective expressions obtained.

M-Polynomial and NM-Polynomial Methods for Topological Indices of Polymers

Topological indices (TIs) are numerical tools widely applied in chemometrics, biomedicine, and bioinformatics for predicting diverse physicochemical attributes and biological activities within molecular structures. Despite their significance, the challenges in deriving TIs necessitate novel approaches. This study addresses the limitations of conventional methods in dealing with dynamic molecular structures, focusing on the neighborhood M-polynomial (NM-polynomial), a pivotal polynomial for calculating degree-based TIs. Current literature acknowledges these polynomials but overlooks their limited adaptability to intricate biopolymer relationships. Our research advances by computing degree-based and neighborhood degree-based indices for prominent biopolymers, including polysaccharides, poly-γ-glutamic acid, and poly-L-lysine. Through innovative utilization of the NM-polynomial and the M-polynomial, we establish a fresh perspective on molecular structure and topological indices. Moreover, we present diverse graph representations highlighting the nuanced correlations between indices and structural parameters. By systematically investigating these indices and their underlying polynomials, our work contributes to predictive modelling in various fields. This exploration sheds light on intricate biochemical systems, offering insights into applications encompassing medicine, the food industry, and wastewater treatment. This research deepens our understanding of complex molecular interactions and paves the way for enhanced applications in diverse industries.

On analysis of iron (II) chloride via graph entropy measures and statistical models.

The crystalline material that is greenish-white and dissolves in water is iron chloride. It is utilized in sewage treatment, dyeing, and medicine. Graph entropy plays a significant role in measuring the complexity of atoms, molecules, and structures in nature. It has specific chemical applications in biology, neuroscience, and chemistry. A compound's molecular structure consists of many atoms. Particularly, hydrocarbons are a chemical combination of hydrogen and carbon atoms. In this article, we discuss the entropy of the chemical structure Iron (II) Chloride. Additionally, we discuss the idea of degree-based indices and compute the Shannon entropy(ENT) using these indices. The linear regression(LR) of various indices and entropies for iron chloride, FeCl2, is also discussed. Also, we link the degree-based indices and entropies via line fit.

Entropy and Multi-Fractal Analysis in Complex Fractal Systems Using Graph Theory

In 1997, Sierpinski graphs, S(n,k), were obtained by Klavzar and Milutinovic. The graph S(1,k) represents the complete graph Kk and S(n,3) is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by Sie(Λ,t), already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph Sie(Λ,t). In the second step, we extract some degree-based indices of a generalized Sierpinski graph Sie(Λ,t) using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph Sie(Λ,t) by using the Randić index.

QSPR Analysis of Some Important Drugs Used in Heart Attack Treatment via Degree-Based Topological Indices and Regression Models

Degree-based topological indices are very useful tools to model and characterize the molecular structure of drugs in order to predict their physicochemical properties without going into laborious and time-consuming laboratory experiments. These indices are numerical descriptors derived for the molecular structures using the principles of graph theory. Degree-based topological indices play a vital role in the QSPR analysis of heart attack drugs by providing molecular descriptors to predict their properties. The main goal of this paper is to compute six degree-based topological indices and a regression model for seven heart attack drugs. These drugs are nitroglycerin, clopidogrel, beta-blockers (metoprolol), ACE inhibitors (lisinopril), statins (atorvastatin), (ARBs) losartan, and beta-adrenergic blockers (propranolol). Regression analysis and degree-based indices correlate with various physicochemical properties related to drug activities, such as molecular weight, complexity, melting point, and boiling point. Correlations provide insights into how the molecular structure influences these properties, helping design and optimize new drugs. In the results, various statistical parameters are used to analyze heart attack drugs.

Relations between p-Sombor and Other Degree-Based Indices

Relations between p-Sombor and Other Degree-Based Indices

QSPR Analysis of Some Phenolic Compounds Present in Moringa Oleifera

QSPR analysis can be used in a variety of applications, such as drug discovery, materials science, and environmental chemistry. For example, QSPR models can be used to predict the toxicity of new chemicals, to design more effective drugs, or to optimize the properties of materials for specific applications. Phenolic compounds act as anti-oxidants, anti-carcinogenic, and anti-bacterial agents present in Moringa oleifera. These Phenolic compounds have significant antioxidant properties as they can scavenge free radicals, slow the aging process, and improve glucose uptake in muscles. They are also used to prevent the accumulation of lipids in the liver and subsequently prevent fatty liver disease, protect kidneys, prevent cancer, prevent whitening skin, chemopreventive for cancer, induce apoptosis in ovarian cancer, protect kidneys, decrease pro-inflammatory enzymes, and increase antioxidants enzymes. Furthermore, phenolic compounds inhibit enzymes linked to diabetes type-2 and they can be used to cure diabetes. Owing to the significance of these phenolic compounds, we carried out QSPR analysis using degree-based indices to know the physical characteristics affiliated with them. The outcomes indicate that these phenolic compounds could be investigated further by pharmacists and chemists when designing drugs using these topological indices values. Since the correlation for the topological indices has been discovered, positively correlated compounds may be considered for the combination of novel drug designs. Also, the computed results are useful to predict the physical properties of the phenolic compounds.

On the construction of some bioconjugate networks and their structural modeling via irregularity topological indices.

Bioconjugate networks refer to networks that are formed by connecting different molecules or particles (such as proteins, enzymes, or nanoparticles) through covalent or non-covalent interactions. These networks are often used in various biological and biomedical applications, such as drug delivery, biosensors, and tissue engineering. The specific properties and behavior of these networks depend on the types of molecules used and the nature of their interactions, which can be studied using various computational and experimental techniques. Farnesyl and geranyl groups are types of isoprenoid chains that are commonly found in various biological molecules such as proteins, lipids, and pigments. The addition of these groups to penicillin molecules may alter their physical and chemical properties, such as solubility, stability, and bioavailability. To gain a better understanding of the structure-property relationships of these antibiotics, this study computes various irregularity indices such as the Albertson index, irregularity index, total irregularity index, Randić irregularity index, and other degree-based indices for two types of sensitive bonds of bioconjugate networks. Numerical results and graphical representations are used to illustrate these findings. The obtained results provide valuable insights into the structure-property relationships of penicillins, which will aid in a better understanding of their behavior and developing more effective antibiotics.

Computation of Entropy Measures for Metal-Organic Frameworks.

Entropy is a thermodynamic function used in chemistry to determine the disorder and irregularities of molecules in a specific system or process. It does this by calculating the possible configurations for each molecule. It is applicable to numerous issues in biology, inorganic and organic chemistry, and other relevant fields. Metal-organic frameworks (MOFs) are a family of molecules that have piqued the curiosity of scientists in recent years. They are extensively researched due to their prospective applications and the increasing amount of information about them. Scientists are constantly discovering novel MOFs, which results in an increasing number of representations every year. Furthermore, new applications for MOFs continue to arise, illustrating the materials' adaptability. This article investigates the characterisation of the metal-organic framework of iron(III) tetra-p-tolyl porphyrin (FeTPyP) and CoBHT (CO) lattice. By constructing these structures with degree-based indices such as the K-Banhatti, redefined Zagreb, and the atom-bond sum connectivity indices, we also employ the information function to compute entropies.

Novel Degree-Based Topological Descriptors of Fenofibrate Using M-Polynomial

Chemical graph theory is currently expanding the use of topological indices to numerically encode chemical structure. The prediction of the characteristics provided by the chemical structure of the molecule is a key feature of these topological indices. The concepts from graph theory are presented in a brief discussion of one of its many applications to chemistry, namely, the use of topological indices in quantitative structure-activity relationship (QSAR) studies and quantitative structure-property relationship (QSPR) studies. This study uses the M-polynomial approach, a newly discovered technique, to determine the topological indices of the medication fenofibrate. With the use of degree-based topological indices, we additionally construct a few novel degree based topological descriptors of fenofibrate structure using M-polynomial. When using M-polynomials in place of degree-based indices, the computation of the topological indices can be completed relatively quickly. The topological indices are also plotted. Using M-polynomial, we compute novel formulas for the modified first Zagreb index, modified second Zagreb index, first and second hyper Zagreb indices, SK index, S K 1 index, S K 2 index, modified Albertson index, redefined first Zagreb index, and degree-based topological indices.