Topological Indices Research Articles

Statistical Evaluation of Cancer Drugs by QSPR Modeling

The prospect of discovering a cure for cancer has been present for the past two to three decades. Annually, this illness affects approximately 10 million individuals worldwide. Anticancer medications are pivotal in treating cancer and other malignant diseases. In this paper, the physical properties and chemical reactions associated with the anticancer medications of various topological indices (TIs) have been established. Additionally, we have discussed the degree-based TIs and their Quantitative Structure-Property Relationship (QSPR) analysis. In mathematical chemistry, molecular descriptors are essential, particularly for researchers investigating Quantitative Structure-Activity Relationship (QSAR) and QSPR models. The goal of the QSPR study is to establish a mathematical relationship between the properties under investigation (such as boiling point and flash point) and various descriptors related to the molecular structure of the drugs. Furthermore, we show the correlation with the physicochemical properties of anticancer medications.

On physical analysis of topological indices for iron disulfide network via curve fitting model

In this study, we conduct a comprehensive physical analysis of topological indices for the Iron Disulfide (FeS2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{2}$$\\end{document}) network using a curve-fitting model. Iron Disulfide is a cubic compound. In metamorphic rock, sedimentary rock, and quartz veins, it is typically found in combination with other sulfides or oxides. The numerical properties of molecular structures are referred to as topological indices. There are several different kinds of topological indices, including those that are based on distance, degree, or counting, among other factors. The real process of creating a topological index involves turning a chemical structure into a numerical value. In this paper, we calculate the iron disulfide network topological indices using the degrees of vertices in a chemical network of Iron Disulfide (FeS2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{2}$$\\end{document}). Thereafter, we discovered the physical parameters of FeS2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{2}$$\\end{document} production, such as heat of formation. We then fitted curves between the thermodynamic properties and several indices. Several techniques based on rationality, linearity, and nonlinearity were used to fit curves in MATLAB. These quantitative results imply that a variety of thermodynamic characteristics of semiconducting materials may be accurately predicted by topological indices. These findings have significant ramifications as they provide the groundwork for the application of topological indices in semiconducting network design and optimization, which might result in more effective and economical material creation.

Face-degree-based topological descriptors of germanium phosphide

In graph theory, topological indices are numerical metrics that give information about a graph’s structural traits. The face index is one such topological index that describes planar networks. Since the discovery of graphene, the genealogy of two-dimensional 2D crystals has expanded and currently contains a large variety that has all logical electrical properties required for nano electronics. Nanotechnology benefits from the use of materials that resemble Dirac, such as silicon, graphite, semiconductors, and germanene, as well as TMDC (phosporene), a transition metal dichalcogenide. In contrast with standard topological descriptors, which are numerical values utilised to characterise molecular structures, the face index presents a potentially more comprehensive method for obtaining structural details. In investigations involving quantitative structure-property relationships (QSPR), this may result in more precise predictions. We calculated the recently created face index of Germanium Phosphide (GeP) and its many shapes, including triangle, rhombus, hourglass, and concentric circles.

Clustering Molecules at a Large Scale: Integrating Spectral Geometry with Deep Learning.

This study conducts an in-depth analysis of clustering small molecules using spectral geometry and deep learning techniques. We applied a spectral geometric approach to convert molecular structures into triangulated meshes and used the Laplace-Beltrami operator to derive significant geometric features. By examining the eigenvectors of these operators, we captured the intrinsic geometric properties of the molecules, aiding their classification and clustering. The research utilized four deep learning methods: Deep Belief Network, Convolutional Autoencoder, Variational Autoencoder, and Adversarial Autoencoder, each paired with k-means clustering at different cluster sizes. Clustering quality was evaluated using the Calinski-Harabasz and Davies-Bouldin indices, Silhouette Score, and standard deviation. Nonparametric tests were used to assess the impact of topological descriptors on clustering outcomes. Our results show that the DBN + k-means combination is the most effective, particularly at lower cluster counts, demonstrating significant sensitivity to structural variations. This study highlights the potential of integrating spectral geometry with deep learning for precise and efficient molecular clustering.

Assessing ecosystem health: A preliminary investigation of the gosikhurd dam ecosystem structure and functioning, an appraisal based on ecological modelling, India.

This study aims to comprehensively understand the Gosikhurd Dam ecosystem (GDE) ecosystem by employing the Ecopath with Ecosim software (version 6.6.5) to construct a trophic mass balancing model. This model consisted of 16 functional groups of organisms, and their interactions and trophic levels were explored. The study focuses on various performance indicators to assess the ecosystem's maturity and complexity. To achieve these objectives, monthly fish samples were collected from June 2022 to May 2023. Performance indicators such as the connectance index (CI), system omnivory index (SOI), Finn's cycling index (FCI), mean path length (FML), ascendency, overhead, and Shannon diversity index were calculated to assess ecosystem maturity and complexity. The Finn's cycling index (FCI) and the mean path length (FML) were calculated as 1.81 and 2.20, respectively, indicating the ecosystem's responsiveness to environmental changes and overall system health and stability. Ascendency and overhead values were also analysed, with ascendency being relatively higher (41.58%), reflecting a system that utilises less than half of its total capacity. The overhead value (58.42%) indicated that the ecosystem is relatively stable and capable of adapting to external perturbations. Furthermore, the Shannon diversity index was 1.67, illustrating less diversity and validating the ecosystem's immaturity. The study identifies critical species and their roles in shaping the ecosystem dynamics, highlighting the importance of zooplankton, zoobenthos, and tilapia as keystone species. These indices propound that GDE is in its developmental stage and lacks complexity compared to mature ecosystems. The findings provide valuable insights into the current state of the ecosystem and can guide future management and conservation efforts.

Reverse degree-based topological indices study of molecular structure in triangular ϒ-graphyne and triangular ϒ-graphyne chain

Topological indices are mathematical descriptors of the structure of a molecule that can be used to predict its properties. They are derived from the graph theory, which describes the topology of a molecule and its connectivity. The main objective is mathematical modeling and topological properties of ϒ-graphyne. Current research focuses on two structures made from hexagonal honeycomb graphite lattices named triangular ϒ-graphyne and triangular ϒ-graphyne chains. The authors have simultaneously computed the first and second Reverse Zagreb indices, reverse hyper-Zagreb indices, and their polynomials. This research also derives mathematical closed-form formulas for some of its fundamental degree-based molecular descriptors. Researchers have been trying to synthesize a novel carbon form called Graphyne. For over a decade but with no success. Recently, some researchers have made a breakthrough in generating Carbons elusive allotrope and solved a long-standing problem in carbon materials. This wonder material is created to rival the conductivity of graphene but with control. These results opened new ways of research in the fields of semiconductors, electronics and optics. Furthermore, graphical and tabular results will help to investigate the structure-property relationships in γ-graphyne.

A Note on Wiener and Hyper-Wiener Indices of Abid-Waheed Graph

A Hosoya polynomial is a polynomial connected to a molecular graph, which is a graph representation of a chemical compound with atoms as vertices and chemical bonds as edges. A graph invariant is the Hosoya polynomial; it is a graph attribute that does not change under graph isomorphism. It provides information about the number of unique non-empty subgraphs in a given graph. A molecular graph's size and branching complexity are determined by a topological metric known as the Wiener index. The Wiener index of each pair of vertices in a molecular network is the sum of those distances. The topological index, one of the various classes of graph invariants, is a real number related to a connected graph's structure .The goal of this article is to compute the Hosoya polynomial of some class of Abid-Waheed graph. Further, this research focused on a C++ algorithm to calculate the wiener index of and . The Wiener index (W∗I) and Hyper-Wiener index (H∗W∗I) are calculated using Hosoya polynomial (H∗-polynomial) of some family of Abid-Waheed graphs and . Illustrations and applications are given to enhance the research work.

Role of Solvents in Oriented Attachment of Ag Nanoparticles: Insights from Molecular Dynamics Simulations and Topological Analysis.

Although the solvation force is considered one of the key forces behind the oriented attachment (OA), the precise roles of solvents in this process remain incompletely elucidated. In this study, we examined the effect of solvent polarities (water, acetone, and chloroform) on the attachment of silver nanoparticles by calculating the free energy curves for the OA process. The observed magnitudes of the binding energies and approaching and dissociation energy barriers are commensurate with the respective solvent polarities. Consequently, OA is more likely to occur in acetone with an intermediate permittivity relative to that of water and chloroform. Additionally, we identified a topological descriptor, namely, the Euler characteristic, of the solvent network, especially the water network, between two approaching surfaces, which manifests a linear correlation with the observed free energy profiles. This descriptor holds promise as a quantitative tool for predicting interactions between nanoparticles in solvent environments featuring hydrogen bond networks.

On physical analysis of phthalocyanine iron (II) using topological descriptor and curve fitting models

A new area of applied chemistry called chemical graph theory uses combinatorial techniques to explain the complex interactions between atoms and bonds in chemical systems. This work investigates the use of edge partitions to decipher molecular connection patterns. The main goal is to use topological indices that capture important topological features to create a connection between the thermodynamic properties and structural characteristics of chemical molecules. We specifically examine the complex web of atoms and links that make up the Fe phthalocyanine chemical graph. Moreover, our study demonstrates a relationship between the calculated topological indices and the thermodynamic properties of Fe phthalocyanine (Phthalocyanine Iron (II)). This work offers insight into the thermodynamic consequences of molecule structures. It advances the subject of chemical graph theory, providing a useful perspective for future applications in catalysis and materials science.

New combined Inverse-QSAR and molecular docking method for scaffold-based drug discovery

Computer-aided drug discovery plays a vital role in developing novel medications for various diseases. The COVID-19 pandemic has heightened the need for innovative approaches to design lead compounds with the potential to become effective drugs. Specifically, designing promising inhibitors of the SARS-CoV-2 main protease (Mpro) is crucial, as it plays a key role in viral replication. Phytochemicals, primarily flavonoids and flavonols from medicinal plants, were screened. Fifty small molecules were selected for molecular docking analysis against SARS-CoV-2 Mpro (PDB ID: 6LU7). Binding energies and interactions were analyzed and compared to those of the anti-SARS-CoV-2 inhibitor Nirmatrelvir. Using these 50 structures as a training set, a QSAR model was built employing simple, reversible topological descriptors. An inverse-QSAR analysis was then performed on 2⁹ = 512 hydroxyl combinations at nine possible positions on the flavone and flavonol scaffold. The model predicted three novel, promising compounds exhibiting the most favorable binding energies (−8.5 kcal/mol) among the 512 possible hydroxyl combinations: 3,6,7,2′,4′-pentahydroxyflavone (PF9), 6,7,2′,4′-tetrahydroxyflavone (PF11), and 3,6,7,4′-tetrahydroxyflavone (PF15). Molecular dynamics (MD) simulations demonstrated the stability of the PF9/Mpro complex over 300 ns of simulation. These predicted structures, reported here for the first time, warrant synthesis and further evaluation of their biological activity through in vitro and in vivo studies.

Correlations between meaning in life and nature connectedness: German-language validation of two topic-related measures and practical implications

Abstract Meaning in life and nature connectedness are indicators of a healthy life and correlate with health and well-being. We conducted a validation study of German versions of the Meaning in Life Questionnaire (Steger et al. 2006) and Nature Connection Index (Richardson et al. 2019) and assessed mindfulness as a trait and personality (big five personality traits) among participating students (n = 353). Results indicated a significant positive correlation between nature connectedness and presence of meaning in life. Presence of meaning correlated significantly with less neuroticism, more openness to new experiences, greater conscientiousness, and moderately with the two mindfulness subscales, ‘acceptance’ and ‘presence’. Search for meaning was associated with more neuroticism and less acceptance. A higher nature-connectedness score correlated with more presence and acceptance (mindfulness), greater openness, and less neuroticism (personality traits). Furthermore, we introduce a nature-assisted model of meaning fulfilment that can be applied in healthcare practices, counselling, and (psycho-) therapeutic treatment. Our results contribute to growing evidence on the preventive and curative effects of nature connectedness and meaning in life and their role as health-related resources. Nature connectedness and meaning in life are effective resources considering various aspects related to (global) crises such as climate change and the corona pandemic.

Exploring Entropy Measures with Topological Indices on Subdivided Cage Networks via Linear Regression Analysis

ABSTRACT In this study, we investigate entropy measurements for subdivided cage networks based on topological indices. We specifically calculate different entropy, redefining Zagreb entropy, HM ( G ) , M 1 ( G ) , M 2 ( G ) entropy, atom bond connection entropy, and Randic entropy. We examine the graphical behavior of various entropy measures using the line fit approach. The results highlight patterns in the distribution of entropy values and interactions between them, which shed light on the intricate connectivity and structural properties of segmented cage networks. This work improves our understanding of cage network dynamics and provides a visual framework for interpreting their behavior.

Topological photonic band gaps in honeycomb atomic arrays

lattice of two-level atoms coupled by the in-plane electromagnetic field may exhibit band gaps that can be opened either by applying an external magnetic field or by breaking the symmetry between the two triangular sublattices of which the honeycomb one is a superposition. We establish the conditions of band gap opening, compute the width of the gap, and characterize its topological property by a topological index (Chern number). The topological nature of the band gap leads to inversion of the population imbalance between the two triangular sublattices for modes with frequencies near band edges. It also prohibits a transition to the trivial limit of infinitely spaced, noninteracting atoms without closing the spectral gap. Surrounding the lattice by a Fabry-Pérot cavity with small intermirror spacing d < \pi/k_0d<π/k0, where k_0k0 is the free-space wave number at the atomic resonance frequency, renders the system Hermitian by suppressing the leakage of energy out of the atomic plane without modifying its topological properties. In contrast, a larger dd allows for propagating optical modes that are built up due to reflections at the cavity mirrors and have frequencies inside the band gap of the free-standing lattice, thus closing the latter.

On curve fitting between topological indices and Gibb’s energy for semiconducting carbon nitrides network

Quantitative structure relationships linked to a chemical structure that shed light on its properties and chemical reactions are called topological indices. This structure is upset by the addition of silicon (Si) doping, which changes the electrical and optical characteristics. In this article, we examine the connection between a chemical structure’s Gibbs energy (GE) and K-Banhatti indices. In this article, we compute the K-Banhatti indices and then show the correlation between the indices and Gibb’s energy of the molecule using curve fitting. Through the curve fitting, we see that there is a strong correlation between indices and Gibb’s energy of a molecule. We use the polynomial curve fitting approach to see the correlation between indices and Gibb’s energy.

Algorithms for Distance-based Topological Indices for Zero Divisor Graphs of Commutative Rings with Primes

Mathematical solutions are sometimes complex to solve mathematically. Algorithms can provide the solution to these problems. Our article introduced algorithms to compute distance-based topological indices for zero divisor graphs containing finite rings as Zp1p2 × Zq. and Zp2 × Zq, having primes p1, p2, and q. Algorithm results can be reused in standing physical structures, solving computer network problems, and designing mechanics.

Group Theoretic Approach towards the Balaban Index of Catacondensed Benzenoid Systems and Linear Chain of Anthracene

In this work, we present the analytical closed forms of the Balaban index for anthracene and catacondensed benzenoid systems using group theoretic techniques. The Balaban index is a distance-based topological index that provides valuable information about the properties of chemical structures. We emphasize the importance of determining analytical closed forms of the Balaban index for catacondensed benzenoid systems and linear chains of anthracene, as it enables a deeper understanding of these systems and their behavior. Our analysis utilizes the group action of the automorphism group of these chains on the set of vertices, which refer to the points where the chains intersect. In future work, we plan to determine the Balaban index of other polymeric linear chains using group theoretic techniques and extend the applications of this index to other fields, such as materials science and biology. It is clear that the Balaban index remains a valuable tool in theoretical and computational chemistry, and its applications are constantly evolving.

Changes in urban green space configuration and connectivity using spatial graph-based metrics in Ardabil developing city, Iran.

Urban planning is essential for managing the diverse impacts of urban green spaces, such as public access, stormwater control, urban life quality, and landscape aesthetics, promoting sustainable urban development and urban residents' well-being by integrating green space considerations into city planning. The aim of this study is to use graph-based metrics to calculate the connectivity of UGS across the main municipal zones of Ardabil city over consecutive periods under different population growth rates. Another objective of this study is to compare the connectivity values of UGS in the four municipal zones and to evaluate changes in the connectivity indices at various distance thresholds of UGS patches. After identifying UGS in different periods, the changes in graph-based connectivity indices at various distance thresholds of UGS patches were analyzed. Additionally, the changes in connectivity indices over different periods and across various municipal zones were compared and analyzed. The findings reveal that UGS areas were larger in the past but have recently had smaller patch sizes. Connectivity between UGS nodes (dNL) decreased at various distances over the study years, showing a declining trend in different connectivity indices. UGS connectivity decreased in municipal zones 1, 2, and 3 but increased in recent years after a decline until 2012 across all four zones of Ardabil city. Zone 4 had the highest UGS connectivity due to newly developed urban areas and well-allocated UGSs. Integrating the ecological impacts of UGS connectivity in urban development and design will enhance trade-offs between conservation, public health, and social equity. New urban areas should allocate sufficient land for UGS and parks, ensuring accessibility to support health and leisure through municipal planning. The study highlights the need for sustainable urban development policies that prioritize the allocation and maintenance of UGSs.

Entropy measures of dendrimers using degree based indices

Topological indices play a crucial role as molecular descriptors in QSAR/QSPR research. Graph entropy measurements are of great importance in various fields like chemistry, discrete mathematics, and biology. Information-theoretic values derived from topological indices, influenced by Shannon’s entropy, are used to analyze the structural characteristics of chemical graphs and complex networks. We discuss the analysis of graph entropies obtained from a new information function. It is equivalent to both the total number of edges and the various degrees of vertices. The information function is also employed to compute the entropies of the system and build a connection between connectivity indices and degrees. This study examines the chemical graphs of Carboxylate-terminated Zinc Phthalocyanine (CtZP) and Nanostar (NS), using the function to connect degree-based topological indices like First Zegrab index, Second Zagreb index, Hyper Zagreb index, Forgotten index, First Redefined Zagreb index, The Second Redefined Zagreb index, The Third Redefined Zagreb index, and Somber index, to calculate the entropies of these structures.

Computing Degree-Based Topological Descriptors of Certain Tessellations of Kekulenes Using M-Polynomial and Neighborhood M-Polynomial

Topological indices are well-known molecular descriptors that may be applied to any graph and to model specific molecular structures, utilized for property correlation in quantitative structure activity and property relationship (QSAR and QSPR) analyses. Strong methods for machine learning information and thermodynamic simulations of massive networks and nanomaterials can be facilitated by using topological indices. Kekulenes are a novel type of cycloarenes, arranged in doughnut-shaped cyclic benzene rings with riveting superaromaticity and other significant properties. The main aim of this paper is to ascertain the notable degree and the neighborhood degree sum-based indices through M-polynomials and neighborhood M-polynomials of certain types of tessellations of kekulenes. Then, a graphical comparison of the resulting indices is also performed.

On Topological Indices of Fuzzy Random Graph

On Topological Indices of Fuzzy Random Graph