Molecular Topological Indices Research Articles

Development and uniqueness test of highly selective atomic topological indices based on the number of attached hydrogen atoms

On the basis of the atomic graph-theoretical index – aEAID (atomic Extended Adjacency matrix IDentification) and molecular adjacent topological index – ATID (Adjacent Topological IDentification) suggested by one of the authors (Zhang Q), a highly selective atomic topological index - aATID (atomic Adjacent Topological IDentification) index was suggested to identify the equivalent atoms in this study. The aATID index of an atom was derived from the number of the attached hydrogen atoms of the atom but omitting bond types. In this case, the suggested index can be used to identify equivalent atoms in chemistry but perhaps not equivalent in the molecular graph. To test the uniqueness of aATID indices, the virtual atomic data sets were derived from alkanes containing 15–20 carbon atoms and the isomers of Octogen, as well as a real data set was derived from the NCI database. Only four pairs of atoms from alkanes containing 20 carbons can't be discriminated by aATID, that is, four pairs of degenerates were found for this data set. To solve this problem, the aATID index was modified by introducing distance factors between atoms, and the 2-aATID index was suggested. Its uniqueness was examined by 5,939,902 atoms derived from alkanes containing 20 carbons and further 16,166,984 atoms from alkanes of 21 carbons, and no degenerates were found. In addition, another large real data set of 16,650,688 atoms derived from the PubChem database was also used to test the uniqueness of both aATID and 2-aATID. As a result, each atom was successfully discriminated by any of the two indices. Finally, the suggested aATID index was applied to the identification of duplicate atoms as data pretreatment for QSPR (Quantitative Structure–Property Relationships) studies.

Supervised Machine Learning-Graph Theory Approach For Analyzing the Electronic Properties of Alkanes

The combination of advanced scientific computing and quantum chemistry improves the existing approach in all chemistry and material science fields. Machine learning has revolutionized numerous disciplines within chemistry and material science. In this study, we present a supervised learning model for predicting the HOMO and LUMO energies of alkanes, which is trained on a database of molecular topological indices. We introduce a new moment topology approach has been introduced as molecular descriptors. Supervised learning utilizes artificial neural networks and support vector machines, taking advantage of the correlation between the molecular descriptors. The result demonstrate that this supervised learning model outperforms other models in predicting the HOMO and LUMO energies of alkanes. Additionally, we emphasize the importance of selecting appropriate descriptors and learning systems, as they play crucial role in accurately modeling molecules with topological orbitals.

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 the chemical characteristics of fullerenes as modelled by R index

A fundamental allotrope of carbon that has drawn significant interest in the fields of nanoscience, nanotechnology, condensed matter physics and chemistry, biological physics, materials science and technology, mechanical and electrical sciences, biomedical engineering, and most recently, medical nanotechnology, and nanoneuroscience. Fullerenes are another important allotrope of carbon in addition to graphite, diamond, and nanotubes. The use of molecular topological indices is crucial for bridging the gap between theoretical and practical aspects of chemical characteristics. In recent years, topological indices have been used to investigate the chemical and physical characteristics of fullerenes. This paper uses the [Formula: see text] molecular topological index to present topological modeling of the binding energies, heats of formation, shape resonances, and Ramsauer-Townsend minima of fullerenes.

Topological Characterization of Hexagonal Network and Non-Kekulean Benzenoid Hydrocarbon

Graph theory has great deal of application in disciplines like biology, computer science, informatics, engineering, mathematics, social science, linguistics and medicine. Topological indices are created by labeling the chemical structure through numerical parameters. These notations link chemical compounds physicochemical features such as boiling temperature, stability, strain energy etc, to its molecular structure. Different topological features of chemical structure have been explored. One of the molecular graphs topological indices are the connection indices defined on the number of connection between vertices of distance two. In this work, we compute the first Zagreb connection index, second Zagreb connection index, and modified first Zagreb connection index for hexagonal network and non kekulean benzenoid hydrocarbon.

On novel topological characteristics of graphene

Molecular topological indices are quantitative measurements that only take into account the topology of a compound’s molecular graph, disregarding any knowledge of the atom locations or chemical connections. An allotrope of carbon called graphene is composed of a single layer of atoms set up in a hexagonal lattice nanostructure. A single layer of carbon atoms organized in a honeycomb pattern make up the two-dimensional substance known as graphene. Graphene possesses significant inherent qualities like strong strength and great thermal and electrical conductivity. It is a sustainable substance with practically countless eco-friendly applications. In this study, we calculate the R molecular topological indices, S molecular topological indices, and Van molecular topological indices of graphene structure, taking into account all conceivable combinations of the number of rows and hexagons. There are high correlations between R, S Van index values and neighbourhood-based entropy values of graphene. This shows that these new indices can be used in QSPR/QSAR studies in chemistry and physics.

Extremal values of VDB topological indices over F-benzenoids with equal number of edges.

The utilization of molecular structure topological indices is currently a standing operating procedure in the structure-property relations research, especially in QSPR/QSAR study. In the past several year, generous molecular topological indices related to some chemical and physical properties of chemical compounds were put forward. Among these topological indices, the VDB topological indices rely only on the vertex degree of chemical molecular graphs. The VDB topological index of an $ n $-order graph $ G $ is defined as TI(G) = \sum\limits_{1\leq i\leq j\leq n-1}m_{ij}\psi_{ij}, $ where $ \{\psi_{ij}\} $ is a set of real numbers, $ m_{ij} $ is the quantity of edges linking an $ i $-vertex and another $ j $-vertex. Numerous famous topological indices are special circumstance of this expression. f-benzenoids are a kind of polycyclic aromatic hydrocarbons, present in large amounts in coal tar. Studying the properties of f-benzenoids via topological indices is a worthy task. In this work the extremum $ TI $ of f-benzenoids with given number of edges were determined. The main idea is to construct f-benzenoids with maximal number of inlets and simultaneously minimal number of hexagons in $ \Gamma_{m} $, where $ \Gamma_{m} $ is the collection of f-benzenoids with exactly $ m $ $ (m\geq19) $ edges. As an application of this result, we give a unified approach of VDB topological indices to predict distinct chemical and physical properties such as the boiling point, $ \pi $-electrom energy, molecular weight and vapour pressure etc. of f-benzenoids with fixed number of edges.

A note on eigenvalue, spectral radius and energy of extended adjacency matrix

The extended adjacency matrix of a graph has been introduced to expand some beneficial molecular topological indices. In the present work, the spectra of certain classes of vertex or edge-transitive graphs are investigated. Also, we study some spectral properties of extended adjacency matrix. Besides, we study the spectral A e x -spread of a graph. In this way, we explore some new bounds for both the smallest and the largest eigenvalues. Finally, we persue the behaviour of A e x -energy of a graph.

Theoretical investigation on the interactions of microplastics with a SARS-CoV-2 RNA fragment and their potential impacts on viral transport and exposure

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) causes the coronavirus disease-19 (COVID-19) pandemic spread across the world and remains difficult to control. Environmental pollution and habitat conditions do facilitate SARS-CoV-2 transmission as well as increase the risk of exposure to SARS-CoV-2. The coexistence of microplastics (MPs) with SARS-CoV-2 affects the viral behavior in the indoor and outdoor environment, and it is essential to study the interactions between MPs and SARS-CoV-2 because they both are ubiquitously present in our environment. To determine the mechanisms underlying the impact of MPs on SARS-CoV-2, we used molecular dynamic simulations to investigate the molecular interactions between five MPs and a SARS-CoV-2 RNA fragment at temperatures ranging from 223 to 310 K in vacuum and in water. We furthermore compared the interactions of MPs and SARS-CoV-2 RNA fragment to the performance of SARS-CoV-1 and Hepatitis B virus (HBV) RNA fragments in interacting with the MPs. The interaction affinity between the MPs and the SARS-CoV-2 RNA fragment was found to be greater than the affinity between the MPs and the SARS-CoV-1 or HBV RNA fragments, independent of the environmental media, temperature, and type of MPs. The mechanisms of the interaction between the MPs and the SARS-CoV-2 RNA fragment involved electrostatic and hydrophobic processes, and the interaction affinity was associated with the inherent structural parameters (i.e., molecular volume, polar surface area, and molecular topological index) of the MPs monomers. Although the evidence on the infectious potential of SARS-CoV-2 RNA is not fully understood, humans are exposed to MPs via their lungs, and the strong interaction with the gene materials of SARS-CoV-2 likely affects the exposure of humans to SARS-CoV-2.

On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider G as a group and H as a non-empty subset of G. The commuting graph C(G,H), has H as the vertex set, where h1,h2∈H are edge connected whenever h1 and h2 commute in G. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of SL(2,C). Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups.

On Wiener Polarity Index and Wiener Index of Certain Triangular Networks

A topological index of graph G is a numerical quantity which describes its topology. If it is applied to the molecular structure of chemical compounds, it reflects the theoretical properties of the chemical compounds. A number of topological indices have been introduced so far by different researchers. The Wiener index is one of the oldest molecular topological indices defined by Wiener. The Wiener index number reflects the index boiling points of alkane molecules. Quantitative structure activity relationships (QSAR) showed that they also describe other quantities including the parameters of its critical point, density, surface tension, viscosity of its liquid phase, and the van der Waals surface area of the molecule. The Wiener polarity index has been introduced by Wiener and known to be related to the cluster coefficient of networks. In this paper, the Wiener polarity index W p G and Wiener index W G of certain triangular networks are computed by using graph-theoretic analysis, combinatorial computing, and vertex-dividing technology.

A note on some distance-based topological indices of circulant network Cn (1, a)

Let Г be a simple connected graph/network with vertex set V(Г) and edge set E(Г). A topological index is a real number associated to Г that characterizes its topology and is invariant under graph automorphism. If υi and υj are vertices in V(Г), the distance between them denoted by d Г (υi , υj ) refers to the length of the shortest path that connects υi , and υj . A topological index is said to be distance-based if its computation involves distance between vertices. Recently, the exact value of some distance-based topological indices namely Wiener, hyper-Wiener, and Schultz molecular topological index of the circulant network Cn (1, a) for a = 2, 3, 4, and 5 were computed. In this paper, we use the breadth-first search method to compute for some distance-based topological indices of the circulant network Cn (1, a) where a = 6 and . We also provide a general formula for the computation of the Wiener, Schultz, and Gutman index of the circulant network Cn (1, a) where .

On Graph theoretic index of Complementary Benzenoids And Macromolecules

Background: A topological index of a graph G is a numerical parameter related to G which characterizes its molecular topology. In the field of QSAR and QSPR research, theoretical properties of the chemical compounds and their molecular topological indices such as distance connectivity indices and degree connectivity indices are used to predict the bioactivity of different molecular compounds. Materials and Methods: Such an approach is different from the traditional QSAR methodology, where one employs selected simpler physico-chemical properties to predict biological activities of molecules. In order to obtain the structure-activity relationships in which theoretical and computational methods are necessary to find appropriate representations of the molecular structure of chemical compounds. These representations are realized through the molecular descriptors. Molecular descriptors are numbers containing structural information derived from the structural representation used for molecules under study. Results: A topological index defined on molecular structure G can be considered as a real valued function f :G→ R+ which maps each durg molecular structure to certain real numbers. Graphene sheets are composed of carbon atoms linked in hexagonal shapes with each carbon atom covalently bonded to three other carbon atoms. Each sheet of graphene is only one atom thick and each graphene sheet is considered a single molecule. Graphene has the same structure of carbon atoms linked in hexagonal shapes to form carbon nanotubes, but graphene is flat rather than cylindrical.. This paper addresses the problem of computing the Wiener , First Zagreb index and Forgotten index of Complementary graphs of graphene sheets, triangular benzenoid graph, circumcoronene molecular graph and nanostar dendrimers. Conclusion: The line graphs were used for modeling amino acid sequences of proteins and of the genetic code. The connected graphs are isomorphic to self complementary graphs. Recently, molecular graphs have proved to be highly useful for drugs activity. Non empirical parameters of chemical structures derived from graph theoretic formalisms are being widely used by many researchers in studies pertaining to molecular design, pharmaceutical drug-design, and environmental hazard assessment of chemicals.

Quantifying some distance topological properties of the non-zero component graph

Several bioactivities of chemical compounds in a molecular graph can be expected by using many topological descriptors. A topological descriptor is a numeric quantity which quantify the topology of a graph. By defining the metric on a graph related with a vector space, we consider this graph in the context of few topological descriptors, and quantify the Wiener index, hyper Wiener index, Reciprocal complimentary Wiener index, Schultz molecular topological index and Harary index. We also provide the graphical comparison of our results to describe the relationship and dependence of these descriptors on the involved parameters.

Differentiation of alkyl radicals: A route through chemical graph theory

AbstractAlkyl radicals play important roles such as intermediary metabolism, cell damage/injury and death leading to potential mutations. The present investigation using chemical graph theory studied two sets of carboxyl radicals, that is, deprotonated (DPro, radical) and protonated (Pro, radical anion) forms of 5‐ethyl heptanoic acid and 5‐ethenyl hept‐6‐enoic acid (Set I) radicals and 6‐ethyl octanoic acid and 6‐ethenyl oct‐7‐enoic acid (Set II) radicals. The study reveals that the largest eigenvalue (LEV) spectra of the adjacency matrix have a unique value, where the spectra increase from DPro to Pro and from single to double bonded alkyl radical structures, thus forming a scoring function for molecular topological indices. This topological index is presented as a measure for molecular connectivity/branching, where the index is used to predict the refractivity of a series of carboxyl radicals. The statistical correlation coefficient obtained for quantitative structure–property relationship between chemical structure (alkyl radical) and its physical property (refractivity) through heat maps are excellent and ranges within 0.97–0.99. It is further discovered that the vector component of the LEV gives an insight to its structural details, where it captures the node with the highest degree along with the important weighted node, that holds the complete structure (i.e., the radical site), in case of the Pro radical structures. Node centrality, which captures the structural makeup, divides DPro radical T‐shaped structures into two subunits for the signal transduction of important biological process like oral toxicity. Size of the largest clusters is also studied, illustrating the parameter to be less sensitive for differentiating the CC double bonds in the Pro radicals. To our knowledge, this is the first study where pattern recognition has been exemplified through the lower‐diagonal‐upper decomposition matrix of the chemical graph that forms a fingerprint signature to differentiate the alkyl radicals. The present study innovatively digitalizes the chemical structures of alkyl radicals that enables the discovery of structure–property relationship reflected by their molecular branching through machine learning.

Computation of the Forgotten Topological Index and Co-Index for Carbon Base Nanomaterial

Molecular topological indices are numerical descriptor of molecular structure obtained via molecular graph G. The forgotten topological index F(G) is a vertex degree based topological index and it can be expressed with where d(u) denotes the degree of u. The forgotten co-index Co-F(G) is defined as the sum of squares of a graph’s vertex degrees which is not adjacent. In this study, the F(G) and Co-F(G) index of HAC5C7, HAC5C6C7, linear [n]-phenylenes and cyclic phenylenes nanostructures are computed for possible works in the properties of molecules such as structure–property relationship. In addition, polynomials of the forgotten index and co-index are found for these nanostructures.

Molecular topological indices-based analysis of thermodynamic properties of graphitic carbon nitride

Graphitic carbon nitride $$({\text {g-C}}_{3}{\text {N}}_{4})$$ has become a new interdisciplinary research hot spot due to its metal-free and visible-light-responsive nature along with alluring electronic band structure and high physicochemical stability. This fascinating semiconductor conjugated polymer is considered a feasible material for harvesting clean and inexhaustible solar energy in the field of environmental remediation and energy applications. Chemical graph theory plays an important role in modeling and designing the chemical structure of complex compounds. The molecular topological indices are the numerical invariants of a molecular graph and are very useful for predicting the reactivity of respective compound. In this paper, we studied the chemical graph of crystalline graphitic carbon nitride, i.e., $$({\text {g-C}}_{3}{\text {N}}_{4})$$ , via degree-based topological indices such as general Randic index, atom bond connectivity index, geometric arithmetic index and Zagreb-type indices. Curve fitting between the calculated thermodynamic properties and topological indices was done together with the numerical and graphical representation of computed results. This mathematical framework of graphitic carbon nitride based on topological indices presents a comprehensive picture of potentially valuable thermodynamic properties such as entropy and enthalpy which could be useful in structural modification of $$({\text {g-C}}_{3}{\text {N}}_{4})$$ for specific application.

Theoretical chemistry study on the toxicity of some polychlorobiphenyl (PCB) compounds using molecular descriptors

This study was carried out to establish, validate and apply a model that relates the toxicity of some PCBs (2,4-dichlorobiphenyl, 2,4’5-trichlorobiphenyl, 2,2’3,4,4’,5’,6-heptachlorobiphenyl, 2,2’,5,5’-tetrachlorobiphenyl and 2,3,3’4,4’-pentachlorobiphenyl) to molecular descriptors (quantum chemical, physicochemical and topological descriptors) through quantitative structure toxicity relationship. The established model reveal that the total molecular energy, electronic energy, core repulsion energy, energy of the highest occupied molecular orbital, energy of the lowest unoccupied molecular orbital, cosmo area, cosmo volume and dipole moment displayed excellent correlation with LD50 values of the PCBs. Some topological parameters (namely molecular topological index, Weiner index, total connectivity and Balaban index) also exhibited excellent correlation with experimentally determined LD50 data. Two equations suitable for estimating theoretical LD50 values of the studied PCBs were derived using quantum chemical and topological descriptors. These equations gave perfect fitness when experiment and theoretical LD50 values were plotted (i.e. R2 value approximated unity) against each other. Perfect correlation was also obtained between theoretical and experimentally derived LD50 values for 2,2’,3,4’,5’,6-hexaxhlorobiphenyl, 2,2’,4,5,5’-pentachlorobiphenyl, 2,3’,4,4’-tetrachlorobiphenyl and 2,2’,4,4’,5,5’-hexachlorobiphenyl, where were PCBs used for model validation. The model was also used to predict the toxicity of other PCBs that lacks accurate experimental data. The study revealed that the reactivity of PCBs can be influence by the number of substituted chlorine atom and by the position of the reactive aromatic carbon. From the results and findings of the study, it was concluded that the applications of quantitative structure-toxicity relation principles to quantum and topological parameters provide excellent avenue for estimating the degree of toxicity of PCBs.

Computing Molecular Topological Descriptors of Polymeric Networks Modeled by Sierpinski Networks

Sierpinski networks constitute an extensively studied class of graphs of fractal nature applicable in topology, mathematics of Tower of Hanoi, computer science, and Chemical graph theory. Recently the properties of Sierpinski networks have been studied like physico-chemical properties, thermodynamic properties, chemical activity, biological activity, metric dimension and degree based topological indices. In this paper, we further extend the study of Sierpinski networks and computed Zagreb types molecular descriptors (molecular topological indices) and their variants.

General Fifth M-Zagreb Polynomials of Benzene Ring Implanted in the P-Type-Surface in 2D Network

Constitutional formulae of molecules are molecular graphs consisting of atoms as vertices and bonds between them represented as edges. The various physical, chemical, and biological properties of molecules are dependent on their molecular structures. The molecular structure is most important, not only to chemists but also to all scientists. The molecular structure descriptors or topological indices of molecules are a mathematical number or a set of selected invariants of matrices that are used to Quantitative Structure-Activity (-Property) Relationships (QSAR/QSPR) studies. In this paper, we computed some new degree-based topological indices of benzene ring implanted in the P-type-surface in the 2D network and its line subdivision of graph.