Topological Indices Research Articles

Topological Data Analysis for Particulate Gels.

Soft gels, formed via the self-assembly of particulate materials, exhibit intricate multiscale structures that provide them with flexibility and resilience when subjected to external stresses. This work combines particle simulations and topological data analysis (TDA) to characterize the complex multiscale structure of soft gels. Our TDA analysis focuses on the use of the Euler characteristic, which is an interpretable and computationally scalable topological descriptor that is combined with filtration operations to obtain information on the geometric (local) and topological (global) structure of soft gels. We reduce the topological information obtained with TDA using principal component analysis (PCA) and show that this provides an informative low-dimensional representation of the gel structure. We use the proposed computational framework to investigate the influence of gel preparation (e.g., quench rate, volume fraction) on soft gel structure and to explore dynamic deformations that emerge under oscillatory shear in various response regimes (linear, nonlinear, and flow). Our analysis provides evidence of the existence of hierarchical structures in soft gels, which are not easily identifiable otherwise. Moreover, our analysis reveals direct correlations between topological changes of the gel structure under deformation and mechanical phenomena distinctive of gel materials, such as stiffening and yielding. In summary, we show that TDA facilitates the mathematical representation, quantification, and analysis of soft gel structures, extending traditional network analysis methods to capture both local and global organization.

Two-Matchings with Respect to the General Sum-Connectivity Index of Trees

A vertex-degree-based topological index φ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by mG,k. Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index φ. This number was denoted by mkG,φ and called the k-matchings of G with respect to the topological index φ. It turns out that m1G,φ=φG, and so for k≥2, the k-matching numbers mkG,φ can be viewed as kth order topological indices which involve both the topological index φ and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices SCα, over the set Tn of trees with n vertices, when α is a real number in the interval −1,0.

Topological indices based VIKOR assisted multi-criteria decision technique for lung disorders.

Lung disorders involve swelling, inflammation, and muscle tightening around the airways, with symptoms such as coughing, wheezing, shortness of breath, and abnormal fluid build-up. The global prevalence of these conditions is rising, highlighting the need for extensive research to alleviate their severity and symptoms. Due to the chronic nature and recurrence of these disorders, the human body often develops immunity and side effects to certain medications. Therefore, developing novel and appropriate drug combinations is crucial. This study analyzes a dataset of lung disorder drugs, characterized by various topological indices. The structures of 16 drugs used to treat lung disorders are correlated with their physical properties using degree-based graph invariants. When considering specific attributes, the VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method provides an optimal ranking for each drug. The QSPR results highlight the effectiveness of this approach in drug prioritization, offering valuable insights for clinical decision-making and drug development. This methodology can enhance the strategic selection of treatments for lung disorders, leading to improved patient care and better resource allocation.

A Study of Differential Topology on the Magnetically Induced Isotropically Averaged Lorentz Force Density of a Few Simple Molecules.

Quantum chemical topology addresses the study of the chemical structure by applying the tools of differential topology to scalar and vector fields obtained by quantum mechanics. Here, the magnetically induced isotropically averaged Lorentz force density was computed and topologically analyzed for 11 small molecules. Critical points (attractors, repellers, and saddles) were determined and trajectories connecting the attractors computed. It is shown that kinds and numbers of the critical points are to some extent transferable in similar molecules. CC bonds of different orders are endowed with critical points of different kinds close to their center. The sum of topological indices of the isolated critical points is influenced by the presence of repellers on the outer part of the molecules.

Ensemble learning and graph topological indices for predicting physical properties of mental disorder drugs

In this paper, we use the ensemble machine learning technique to evaluate the strength of three supervised machine learning algorithms, namely, the random forest regression (RFR), support vector regression (SVR) and the gradient boosting regression (GBR) in the prediction of physical properties of mental disorder drugs with small dataset. The model was implemented on a dataset of neighborhood degree-based topological indices which served as predictor variables and physical properties of the drugs which served as target variables. To compute the neighborhood degree-based indices, we employed an algorithm that utilizes the canonical SmilES notations of the drugs. The ensemble method identifies the neighborhood third Zagreb index (NM3(G)) as an efficient predictor of boiling point, flash point and enthalpy of vaporization. The neighborhood Randic index (NR(G)) provides better prediction for molar refractivity, molar volume and polarizability. In the same vein, the neighborhood sum connectivity index (NSC(G)) is an efficient predictor of surface tension while the neighborhood reciprocal Randic index (NRR(G)) is most effective in the prediction of polar surface area. Furthermore, the comparison of the average performance between the ensemble method and the base models (RFR, SVR, GBR) over the neighborhood topological indices shows efficient performance of the individual models across multiple physical properties of mental disorder drugs, when using the neighborhood topological indices as the predictor or input feature. Overall, this research highlights the combination of three supervised machine learning models in an ensemble environment to mitigating the challenges associated with small datasets when applying machine learning models in QSPR analysis.

Eccentric connectivity index in complementary prisms

The topological index is just one of several very useful tools that graph theory has made available to chemists. Topological indices are invariants of real numbers under graph isomorphisms. Several topological indices have been defined. Some of them are used to model chemical, pharmaceutical and other properties of molecules. The eccentric connectivity index (eci) is also a topological index. The eci of [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text], where [Formula: see text] represents the degree of a vertex [Formula: see text] and [Formula: see text] is its eccentricity. In this paper, exact formula for the eci of complementary prisms is derived.

A graph theoretical study of porphyrin-cored dendrimers by means of Sombor indices:a computational approach

Topological indices are numerical parameters calculated on the molecular graph of chemical structures which do not depend on the method the molecular graph is labelled or depicted and have no need of energy minimisation of the molecular structure. The Sombor index is a recently established topological index possessing the ability to predict the enthalpy of vaporisation and entropy of octane isomers. Here, we apply the edge-partition technique to study the Sombor index and some of its variants namely delta Sombor index, reduced Sombor index, increased Sombor index, modified Sombor index, reverse Sombor index and Kulli-Gutman Sombor index for four types of dendrimers with porphyrin core. Our results could be helpful in better understanding the physico-chemical characteristics of this family of macromolecules.

Extremal Trees for Logarithmic VDB Topological Indices

Vertex-degree-based (VDB) topological indices have been applied in the study of molecular structures and chemical properties. At present, the exponential VDB index has been studied extensively. Naturally, we began to consider the logarithmic VDB index lnTf. In this paper, we first discuss the necessity of a logarithmic VDB index, and then present sufficient conditions so that Pn and Sn are the only trees with the smallest and greatest values of lnTf(T). As applications, the minimal and maximal trees of some logarithmic VDB indices are determined. Through our work, we found that the logarithmic VDB index lnTf has excellent discriminability, but the relevant results are not completely opposite to the exponential VDB index. The study of logarithmic VDB indices is an interesting but difficult task that requires further resolution.

Spectrum‐Based Topological Indices and Their QSPR Studies of Nonsteroidal Anti‐Inflammatory Drugs

ABSTRACTSpectrum‐based topological indices (eigenvalue‐based topological indices), a valuable tool for analyzing molecular structure. These topological indices are metrics that reflect the inherent characteristics of chemical substances, were employed in conjunction with quantitative structure‐property relationship (QSPR) to investigate nonsteroidal anti‐inflammatory drugs (NSAIDs) which are used toalleviate or eliminate pain sensations in affected areas. We have to use mathematica to compute various eigenvalue‐based indices and utilized statistical software to identify correlations between these indices and key physical properties of NSAIDs. The analysis revealed that specific indices, including index, index, index, and index exhibited strong associations with properties like complexity and refractivity, boiling point, polarity, and molar weight, respectively.

Exploring the role of topological descriptors to predict physicochemical properties of anti-HIV drugs by using supervised machine learning algorithms

In order to explore the role of topological indices for predicting physio-chemical properties of anti-HIV drugs, this research uses python program-based algorithms to compute topological indices as well as machine learning algorithms. Degree-based topological indices are calculated using Python algorithm, providing important information about the structural behavior of drugs that are essential to their anti-HIV effectiveness. Furthermore, machine learning algorithms analyze the physio-chemical properties that correspond to anti-HIV activities, making use of their ability to identify complex trends in large, convoluted datasets. In addition to improving our comprehension of the links between molecular structure and effectiveness, the collaboration between machine learning and QSPR research further highlights the potential of computational approaches in drug discovery. This work reveals the mechanisms underlying anti-HIV effectiveness, which paves the way for the development of more potent anti-HIV drugs. This work reveals the mechanisms underlying anti-HIV efficiency, which paves the way for the development of more potent anti-HIV drugs which demonstrates the invaluable advantages of machine learning in assessing drug properties by clarifying the biological processes underlying anti-HIV behavior, which paves the way for the design and development of more effective anti-HIV drugs.

Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing.

Scalar field comparison is a fundamental task in scientific visualization. In topological data analysis, we compare topological descriptors of scalar fields-such as persistence diagrams and merge trees-because they provide succinct and robust abstract representations. Several similarity measures for topological descriptors seem to be both asymptotically and practically efficient with polynomial time algorithms, but they do not scale well when handling large-scale, time-varying scientific data and ensembles. In this paper, we propose a new framework to facilitate the comparative analysis of merge trees, inspired by tools from locality sensitive hashing (LSH). LSH hashes similar objects into the same hash buckets with high probability. We propose two new similarity measures for merge trees that can be computed via LSH, using new extensions to Recursive MinHash and subpath signature, respectively. Our similarity measures are extremely efficient to compute and closely resemble the results of existing measures such as merge tree edit distance or geometric interleaving distance. Our experiments demonstrate the utility of our LSH framework in applications such as shape matching, clustering, key event detection, and ensemble summarization.

Associations of threat and deprivation-related childhood exposures with children’s mental health and flourishing: The moderating role of family resilience

IntroductionThe association of adverse childhood experience (ACE) with physical and mental well-being is predominantly examined using a cumulative ACE score. However, less is known how different dimensions of adversity (e.g., threat and deprivation) may uniquely predict children’s mental health and development. We examined (i) whether two aspects of ACE – threat and deprivation – are differently associated with children’s anxiety, depression, and flourishing and (ii) whether family resilience and connection index (FRCI) moderates these associations. MethodsWe analyzed data from the National Survey of Children’s Health (2017–20). The analytic sample consisted of 81,626 children (6–17 years). Logistic regression with survey weights was used to obtain odds ratios (OR) and 95 % confidence intervals (CI) adjusting for sociodemographic factors. ResultsThe experience of threat-only ACE and deprivation-only ACE showed differential associations with anxiety, depression, and child flourishing. For example, children who experienced threat-only ACE had a 4.2-fold higher odds of depression (OR=4.2; 95 % CI: 3.37–5.24), and those who experienced deprivation-only ACE had a 2.4-fold (OR=2.4; 95 % CI: 1.98–2.92) higher odds of depression compared to those exposed to neither threat nor deprivation experience. Experiencing both a threat and deprivation showed larger than an additive association with all outcomes. Further, interaction models suggest that the associations of threat and deprivation-type ACE with all outcomes significantly differed by the levels of FRCI. ConclusionsThreat and deprivation dimensions of adversity were differentially associated with mental health and flourishing outcomes in childhood. Family resilience appears to be protective against the negative effect of threat and deprivation dimensions of ACE on children’s mental health and developmental outcomes.

Hydrological Connectivity Response of Typical Soil and Water Conservation Measures Based on SIMulated Water Erosion Model: A Case Study of Tongshuang Watershed in the Black Soil Region of Northeast China

The black soil region of Northeast China is the largest commercial grain production base in China, accounting for about 25% of the total in China. In this region, the water erosion is prominent, which seriously threatens China’s food security. It is of great significance to effectively identify the erosion-prone points for the prevention and control of soil erosion on the slope of the black soil region in Northeast China. This article takes the Tongshuang small watershed (Heilongjiang Province in China) as an example, which is dominated by hilly landforms with mainly black soil and terraces planted with corn and soybeans. Based on the 2.5 cm resolution Digital Elevation Model (DEM) reconstructed by unmanned aerial vehicles (UAVs), we explore the optimal resolution for hydrological simulation research on sloping farmland in the black soil region of Northeast China and explore the critical water depth at which erosion damage occurs in ridges on this basis. The results show that the following: (1) Compared with the 2 m resolution DEM, the interpretation accuracy of field roads, wasteland, damaged points, ridges and cultivated land at the 0.2 m resolution is increased by 4.55–27.94%, which is the best resolution in the study region. (2) When the water depth is between 0.335 and 0.359 m, there is a potential erosion risk of ridges. When the average water depth per unit length is between 0.0040 and 0.0045, the ridge is in the critical range for its breaking, and when the average water depth per unit length is less than the critical range, ridge erosion damage occurs. (3) When local erosion damage occurs, the connectivity will change abruptly, and the remarkable change in the index of connectivity (IC) can provide a reference for predicting erosion damage.

Unveiling correlated two-dimensional topological insulators through fermionic tensor network states—classification, edge theories and variational wavefunctions

The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations andnon-chiraltopological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

Harnessing topological machine learning in Raman spectroscopy: Perspectives for Alzheimer’s disease detection via cerebrospinal fluid analysis

The cerebrospinal fluid of 21 subjects who received a clinical diagnosis of Alzheimer’s disease (AD) as well as of 22 pathological controls has been collected and analysed by Raman spectroscopy (RS). We investigated whether the Raman spectra could be used to distinguish AD from controls, after a preprocessing procedure. We applied machine learning to a set of topological descriptors extracted from the spectra, achieving a high classification accuracy of 86%. Our experimentation indicates that RS and topological analysis may be a reliable and effective combination to confirm or disprove a clinical diagnosis of Alzheimer’s disease. The following steps will aim at leveraging the intrinsic interpretability of the topological data analysis to characterize the AD subtypes, e.g. by identifying the bands of the Raman spectrum relevant for AD detection, possibly increasing and/or confirming the knowledge about the precise molecular events and biological pathways behind the Alzheimer’s disease.

On statistical evaluation of reverse degree based topological indices for iron telluride networks

In the context of graph theory and chemical graph theory, this research conducts a detailed mathematical investigation of reverse topological indices as they relate to iron telluride networks, clarifying their complex interactions. Graph theory is a branch of abstract mathematics that carefully studies the connections and structural features of graphs made up of edges and vertices. These theoretical ideas are expanded upon in chemical graph theory, which models molecular architectures with atoms acting as vertices and chemical bonds as edges. By extending these concepts, this work investigates the reverse topological indices in the context of Iron Telluride networks and outlines their significant effects on chemical reactivity, molecular topology and statistical modeling. By navigating intricate mathematical formalisms and algorithmic approaches, the analysis provides profound insights into the reactivity patterns and structural dynamics of Iron Telluride compounds, enhancing our knowledge of solid-state chemistry and materials science.

Forgotten Topological Index and its Properties on Neutrosophic Graphs

Topological indices play a significant role in crisp, fuzzy graphs and their real-life application. But to avoid the vagueness in the final result, these indices should be dealt with in the neutrosophic environment since it consolidates the uncertain quantity or values of an event in the name of "indeterminacy membership". Except for the wiener index, no other indices are introduced under the neutrosophic graphical setting. In this article, we consider the forgotten topological index (ToI) and the Edge forgotten index in the 3-valued logic neutrosophic graph and came up with some important theorem results and applications.

NM-polynomial-based topological indices and graph entropies of porphyrazine

NM-polynomial-based topological indices and graph entropies of porphyrazine

Modern chemical graph theory

AbstractGraph theory has a long history in chemistry. Yet as the breadth and variety of chemical data is rapidly changing, so too do graph encoding methods and analyses that yield qualitative and quantitative insights. Using illustrative cases within a basic mathematical framework, we showcase modern chemical graph theory's utility in Chemists' analysis and model development toolkit. The encoding of both experimental and simulation data is discussed at various levels of granularity of information. This is followed by a discussion of the two major classes of graph theoretical analyses: identifying connectivity patterns and partitioning methods. Measures, metrics, descriptors, and topological indices are then introduced with an emphasis upon enhancing interpretability and incorporation into physical models. Challenging data cases are described that include strategies for studying time dependence. Throughout, we incorporate recent advancements in computer science and applied mathematics that are propelling chemical graph theory into new domains of chemical study.This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Structure and Mechanism > Computational Materials Science Structure and Mechanism > Molecular Structures

A new understanding of coronary curvature and haemodynamic impact on the course of plaque onset and progression

The strong link between atherosclerosis and luminal biomechanical stresses is well established. Yet, this understanding has not translated into preventative coronary diagnostic imaging, particularly due to the under-explored role of coronary anatomy and haemodynamics in plaque onset, which we aim to address with this work. The left coronary trees of 20 non-stenosed (%diameter stenosis [%DS] = 0), 12 moderately (0 < %DS < 70) and 7 severely (%DS ≥ 70) stenosed cases were dissected into bifurcating and non-bifurcating segments for whole-tree and segment-specific comparisons, correlating nine three-dimensional coronary anatomical features, topological shear variation index (TSVI) and luminal areas subject to low time-average endothelial shear stress (%LowTAESS), high oscillatory shear index (%HighOSI) and high relative residence time (%HighRRT). We found that TSVI is the only metric consistently differing between non-stenosed and stenosed cases across the whole tree, bifurcating and non-bifurcating segments ( p < 0.002, AUC = 0.876), whereas average curvature and %HighOSI differed only for the whole trees ( p < 0.024) and non-bifurcating segments ( p < 0.027), with AUC > 0.711. Coronary trees with moderate or severe stenoses differed only in %LowTAESS ( p = 0.009) and %HighRRT ( p = 0.012). This suggests TSVI, curvature and %HighOSI are potential factors driving plaque onset, with greater predictive performance than the previously recognized %LowTAESS and %HighRRT, which appears to play a role in plaque progression.