Topological Indices Research Articles

Bounds for the Gutman–Milovanović index and some applications

AbstractIn this paper, we examine the Gutman–Milovanović index and establish new upper and lower bounds for it. These bounds include terms related to the general sum connectivity index, the general second Zagreb index, and the hyperbolicity constant of the underlying graph. Also, we model physicochemical properties of polyaromatic hydrocarbons using the Gutman–Milovanović index.

Investigating The Characteristics Of Anti-Malarial Drugs By Cordial Label-Based Topological Indices

Investigating The Characteristics Of Anti-Malarial Drugs By Cordial Label-Based Topological Indices

The Second Hyper-Zagreb Index and Coindex of Chemical Nanostructures in Drugs.

Various dendrimer nanoparticles have properties like multivalency, controlled size, and surface functionality that make them promising nanocarriers for targeted drug delivery and other applications in pharmaceutical sciences. The precise tunability of dendrimers is an advantage over other nanoparticles. The topological descriptors can be used to predict the physicochemical properties of dendrimers and optimize their branching pattern for specific applications. The second hyper-Zagreb index and co-index are computed for various chemical structures, including dendrimers, to facilitate the correlation between their structure and biological activity. In this study, the second Hyper-Zagreb index and second Hyper-Zagreb polynomials were calculated for various chemical structures, such as the molecular graph of poly(propyl) ether imine dendrimer PETIM, nanostar dendrimer (D3[p]), polypropylenimine octaamine dendrimer (NS1[p]) and (NS2[p]), polymer dendrimer (NS3[p]) and NS5[p]), fullerene dendrimer (NS4[p]), and other classes of dendrimers. By computing formulae and analyzing data and figures, we obtained new insights into the features of structure-property connections for these types of compounds of nanostar dendrimers. The results can be used to optimize the properties of dendrimers for specific applications.

On the Non-Commuting Graph of the Group U6n

Let G be a finite group. The non-commuting graph of G is a simple graph Γ(G) whose vertices are elements of G∖Z(G), where Z(G) is the center of G, and two distinct vertices a and b are joint by an edge if ab≠ba. In this paper, we study the non-commuting graph of the group U6n. The independent number, clique and chromatic numbers of the non-commuting graph of the group U6n, Γ(U6n), are determined. Additionally, the resolving polynomial, total eccentricity and independent polynomials of Γ(U6n) are computed. Finally, the detour and eccentric connectivity indices of Γ(U6n) are found.

Returns to workplace connectivity: evidence from Brazil

ABSTRACT We empirically investigate the quantitative effect of workplace connectivity on workers’ wages. Connectivity measures a worker’s rank within the workplace’s connection hierarchy, considering the number of co-workers in the same occupation and firm (workplace) and the relative tenure of immediate co-workers. We construct a connectivity index as the ratio of a worker’s tenure-weighted connections to the total number of connections in the workplace, using a worker’s job tenure and the connections formed over time to obtain a relative measurement of their connectivity. Using a unique dataset from the Brazilian Annual Social Information Report (RAIS), we estimate the (fixed effect) wage equation, controlling for work and firm heterogeneity. Our results show positive and statistically significant returns to connectivity. The effect is particularly pronounced for workers in administrative services and industrial goods production, those at the bottom of the connectivity distribution, and those in medium-sized firms. We also find that a worker’s connectivity is more relevant for wages than their tenure.

Computing Zagreb Connection Indices for the Cartesian Product of Path and Cycle Graphs

Topological indices ( ) are a class of graph-based descriptors widely used in chemiformatics and Quantitative Structure-Activity Relationship (QSAR) studies. TIs capture key structural features of molecules by encoding graph-theoretic information, offering a quantitative representation of molecular topology. Each mathematical network is associated with a specific numerical value determined by a function. Among the Zagreb connection indices have been extensively researched. This study delves into the Cartesian product of cycle graphs with path graphs, elucidating the comprehensive implications of . These indices encompass the first , second , and third . Furthermore, comprehensive results of the modified first , modified second , and modified third are presented, along with the first multiplicative ZCI second multiplicative ZCI (SMZCI), and third multiplicative ZCI Moreover, modified first multiplicative ZCI ), modified second multiplicative ZCI and modified third multiplicative ZCI ( are also calculated. To provide precision, both the graphical and numerical analyses of the computed findings are aligned for the two Cartesian products. The foundation for mathematically modeling complex networks and chemical structures lies in graph theory. Topological indices ( ) are a class of graph-based descriptors widely used in chemiformatics and quantitative structure-activity relationship (QSAR) studies. TIs capture key structural features of molecules by encoding graph-theoretic information, offering a quantitative representation of molecular topology. Each mathematical network is associated with a specific numerical value determined by a topological index function. Among these the Zagreb connection indices are extensively researched TIs. This article delves into the Cartesian product of cycle graphs with path graphs, elucidating the comprehensive implications of . These indices encompass the first, Second and third. Furthermore, we present the comprehensive results of the modified , modified and modified , and also present the first multiplicative Zagreb connection index , and , along with their modified counterparts like modified first multiplicative Zagreb connection index , and . These analysis encompass two distinct types of graphs: cycle graphs and path graphs. To provide further precision, we align both the graphical and numerical analysis of our computed findings for these two Cartesian products.

Are the ecological benefits of hydrological connectivity projects continuously increasing? Insights from the perspective of landscape patterns − ecosystem processes − ecosystem services

The construction of hydrological connectivity projects alters the natural morphology and hydrological rhythms of the basin, which can subsequently impact ecological processes and ecosystem services. Quantifying the impacts of hydrological connectivity projects on ecosystems is fundamental for providing theoretical support to researchers and policy makers. Ecological water conveyance projects (EWCP) aimed at ecological restoration are one type of hydrological connectivity projects. In this study, a multi-dimensional hydrological connectivity assessment framework is established to quantify the impacts of the EWCP on landscape patterns. Analytic Hierarchy Process − Entropy Method (AHP-EM) is used to calculate the integrated hydrological connectivity index (IHCI). Ecosystem processes are characterized by vegetation indexes and ecological indexes. The ecosystem services are quantified by ecosystem service value (ESV). An improved ESV calculation method is proposed to assess the ecological benefits of EWCP. This method not only captures the spatial heterogeneity information of land use, vegetation and ecological conditions but also integrates socioeconomic factors. The relationships between trade-offs and synergies among provisioning, regulating, supporting and cultural ecosystem services are clarified to optimize the overall benefits. The results show that hydrological connectivity increases consistently with ecological water conveyance volume (EWCV). ESV is increased positively with IHCI and it shows a correlation with the distance to the riverbank. A new finding is that the ecological benefit shows an inflection point in 2010, which conforms to the “marginal benefit decreasing law”. These results provide a new insight on hydrological connectivity dynamics under the influence of EWCP, and the finding has guiding significance for efficient utilization of water resources in EWCP. The methodologies used in this research can be widely applied to other EWCPs.

Modified reverse degree descriptors for combined topological and entropy characterizations of 2D metal organic frameworks: applications in graph energy prediction.

Topological descriptors are widely utilized as graph theoretical measures for evaluating the physicochemical properties of organic frameworks by examining their molecular structures. Our current research validates the usage of topological descriptors in studying frameworks such as metal-butylated hydroxytoluene, NH-substituted coronene transition metal, transition metal-phthalocyanine, and conductive metal-octa amino phthalocyanine. These metal organic frameworks are crucial in nanoscale research for their porosity, adaptability, and conductivity, making them essential for advanced materials and modern technology. In this study, we provide the topological and entropy characterizations of these frameworks by employing robust reverse degree based descriptors, which offer insightful information on structural complexities. This structural information is applied to predict the graph energy of the considered metal organic frameworks using statistical regression models.

The Sombor Indices of Banana Tree Graph and Fractal Tree Type Dendrimer

Topological descriptors are graph-theoretical metrics that allow one to describe a molecular structure’s underlying connectedness. Degree-based topological descriptors have been investigated widely and are associated with several chemical characteristics. The examination of graph entropy indices as a gauge for the intricacy of the underlying connection and as a means of characterizing structural attributes has also gained significance.The present study utilizes the graph-theory-based edge partition method to examine Sombor variants for the Banana Tree Graph, Fractal, and Cayley Tree Type Dendrimers. We have employed Shannon’s entropy model to determine the graph-based entropy of these graphs and produced Sombor indices.

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.