Topological Indices Research Articles

Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is ΩYα(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)+dΩ(ϱ)dΩ(ϱ))α, while the second form of the GPSCI is defined as Yβ(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)×dΩ(ϱ)dΩ(ϱ))β. In this paper, we investigate Yβ in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for Yβ using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered.

On the Exponential Atom-Bond Connectivity Index of Graphs

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: eABC=eABC(Υ)=∑vivj∈E(Υ)edi+dj−2didj, where di is the degree of the vertex vi in Υ. In this paper, we prove that the double star DSn−3,1 is the second maximal graph with respect to the eABC index of trees of order n. We give an upper bound on eABC of unicyclic graphs of order n and characterize the maximal graphs. The graph K1∨(P3∪(n−4)K1) gives the maximal graph with respect to the eABC index of bicyclic graphs of order n. We present several relations between eABC(Υ) and ABC(Υ) of graph Υ. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research.

Adjacency spectrum and Wiener index of essential ideal graph of a finite commutative ring

Let R be a commutative ring with unity. The essential ideal graph ER of R, is a graph with a vertex set consisting of all nonzero proper ideals of R and two vertices I and K are adjacent if and only if I + K is an essential ideal. In this paper, we study the adjacency spectrum of the essential ideal graph of the finite commutative ring Zn, for n = {pm, pm1qm2}, where p, q are distinct primes, and m, m1 , m2 ε N. We show that 0 is an eigenvalue of the adjacency matrix of EZn if and only if either n = p2 or n is not a product of distinct primes. We also determine all the eigenvalues of the adjacency matrix of EZn whenever n is a product of three or four distinct primes. Moreover, we calculate the topological indices, namely the Wiener index and hyper-Wiener index of the essential ideal graph of Zn, for different forms of n.

THE NON-COPRIME GRAPHS OF UPPER UNITRIANGULAR MATRIX GROUPS OVER THE RING OF INTEGER MODULO WITH PRIME ORDER AND THEIR TOPOLOGICAL INDICES

In its application graph theory is widely applied in various fields of science, including scheduling, transportation, industry, and structural chemistry, such as topological indexes. The study of graph theory is also widely applied as a form of representation of algebraic structures, including groups. One form of graph representation that has been studied is non-coprime graphs. The upper unitriangular matrix group is a form of group that can be represented in graph form. This group consists of upper unitriangular matrices, which are a special form of upper triangular matrix with entries in a ring R and all main diagonal entries have a value of one. In this research, we look for the form of a non-coprime graph from the upper unitriangular matrix group over a ring of prime modulo integers and several topological indexes, namely the Harmonic index, Wiener index, Harary index, and First Zagreb index. The findings of this research indicate that the structure of the graph and the general formula for the Harmonic index, Wiener index, Harary index, and First Zagreb index were successfully obtained.

Exploration of the Mountainous Urban Rail Transit Resilience Under Extreme Rainfalls: A Case Study in Chongqing, China

Extreme rainfalls could greatly affect operations of urban rail transit systems of mountainous cities, which are prone to have landslides and floods under rainfalls. Therefore, it is essential to assess and enhance the resilience of mountainous urban rail transit networks under heavy rainfalls. Taking the metro network of Chongqing, the largest mountainous city in China, as an example, this study establishes a network topology model to identify the high-risk nodes under rainfalls and find the effective recovery strategies. By introducing the metro ridership and topological shortest distances, a network service efficiency function is developed, and the importance of nodes is quantified using service efficiency index and topological importance index. The resilience assessment model based on service efficiency is constructed using the resilience triangle theory. Additionally, risk levels for landslide and flood-prone areas are classified using the K-means algorithm, based on rainfall, elevation, and slope data, identifying high-risk stations. Finally, the node recovery sequence and strategies for high-risk nodes affected by landslides and floods are examined. The results indicate that in extreme rainfall scenarios, two transfer stations (Daping and Fuhua Road) are among the high-risk landslide stations, while most other nodes have a service efficiency index of less than 0.2. High-risk flood stations are located on non-transfer lines and mostly on metro lines with high traffic flow, with service efficiency index generally high, with some stations, like Bijin Station, exceeding 0.3. When all affected nodes fail, network service efficiency decreases by 84.0% and 75.2% under landslide and flood disasters, respectively. Compared with the random recovery strategy, recovery strategies based on topological importance and service efficiency index, the optimal recovery strategy based on genetic algorithm performs much better.

A python approach for prediction of physicochemical properties of anti-arrhythmia drugs using topological descriptors

In recent years, machine learning has gained substantial attention for its ability to predict complex chemical and biological properties, including those of pharmaceutical compounds. This study proposes a machine learning-based quantitative structure-property relationship (QSPR) model for predicting the physicochemical properties of anti-arrhythmia drugs using topological descriptors. Anti-arrhythmic drug development is challenging due to the complex relationship between chemical structure and drug efficacy. To address this, we applied machine learning techniques, specifically linear regression models combined with K-fold cross-validation, to predict critical properties such as Density, Boiling Point, Flash Point, Bioconcentration Factor (BCF), Organic Carbon Partition Coefficient (KOC), Polarizability, and Molar Volume. The models were developed using data from ten anti-arrhythmic drugs ( to ). We evaluated the models based on performance metrics such as R and and obtained significant results. Most accurate predictions are obtained for polarizability from models with H(G) and .

Chemical Bond Overlap Descriptors From Multiconfiguration Wavefunctions.

The chemical bond is a fundamental concept in chemistry, and various models and descriptors have evolved since the advent of quantum mechanics. This study extends the overlap density and its topological descriptors (OP/TOP) to multiconfigurational wavefunctions. We discuss a comparative analysis of OP/TOP descriptors using CASSCF and DCD-CAS(2) wavefunctions for a diverse range of molecular systems, including X-O bonds in X-OH (XH, Li, Na, H2B, H3C, H2N, HO, F) and Li-X' (XF, Cl, and Br). Results show that OP/TOP aligns with bonding models like the quantum theory of atoms in molecules (QTAIM) and local vibrational modes theory, revealing insights such as overlap densities shifting towards the more electronegative atom in polar bonds. The Li-F dissociation profile using OP/TOP descriptors demonstrated sensitivity to ionic/neutral inversion during Li-F dissociation, highlighting their potential for elucidating complex bond phenomena and offering new avenues for understanding multiconfigurational chemical bond dynamics.

Mathematical modeling and QSPR analysis of hepatitis treatment drugs through connection indices an innovative approach.

Chemical structures may be defined based on their topology, which allows for the organization of molecules and the representation of new structures with specific properties. We use topological indices, which are precise numerical measurements independent of structure, to measure the bonding arrangement of a chemical network. An essential objective of studying topological indices is to collect and alter chemical structure data to develop a mathematical relationship between structures and physico-chemical properties, bio-activities, and associated experimental factors. Topological indices establish the correlation between molecular characteristics, such as physical, chemical, thermodynamic, and biological activity, and their corresponding chemical structures in Quantitative Structure-Property Relationships (QSPR) and Quantitative Structure-Activity Relationships (QSAR). Hepatitis, in its advanced stages, can lead to the development of mental illnesses. Effective management of symptoms frequently entails regular use of medicine and therapy for a prolonged period. Assurance of the safety and efficacy of drug design is of utmost importance. Various parameters, including solubility, metabolic stability, toxicity, permeability, and transporter effects, depending on the physical and chemical properties of the treatment, impact the efficacy of biopharmacological design. Increasingly, computational approaches have become indispensable in the discipline of pharmaceutical discoveries and development. The major objective of this work is to examine the chemical appropriateness of connection indices in evaluating six physico-chemical properties of the 14 medicines used for the treatment of hepatitis. This work conducts QSPR analysis to obtain the most accurate approximations for the physico-chemical attributes of pharmacological agents employed in the treatment of hepatitis.

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.

The role of river connectivity in the distribution of fish in an anthropized watershed.

The ongoing biodiversity crisis is especially severe in freshwater habitats. Anthropized watersheds, such as the Seine-Normandie basin in France, are particularly affected by human interference. The study of fish species distribution in watersheds often relies on environmental drivers such as land use or climate. Yet, fish are also exposed to river connectivity constraints, such as dams, that are understudied despite their potential impact on fish dispersal. For this study, we investigated the role of local and whole-basin longitudinal connectivity in fish distribution. We designed connectivity indices based on river network characteristics and specific mobility for 33 species and included these indices in species distribution models, taking into account habitat suitability, to quantify their role in species distribution. Keeping the best index for each species, an average of 29% - and up to 57% - of explained fish distribution, depending on species, was tied to connectivity. We found that high connectivity often had a significant and positive linear effect on species presence probability. Using a scoring system across multiple indices, we found connectivity indices that took local context into account (e.g. the ecological zonation of the river) performed consistently better than others. Indices that took only dispersal limitation into account scored higher for 12 species, while barriers, alone, were the most important constraint for 10 species, the remaining 11 being associated with both. This work points to fragmentation as a cause for lower likelihood of presence for many non-diadromous river fish species. It highlights the importance of considering both physical and functional connectivity constraints in fish distribution and provides additional insights for river management and restoration.

Uncertainty Visualization of Critical Points of 2D Scalar Fields for Parametric and Nonparametric Probabilistic Models.

This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.

Computation of Differential, Integral Operators and Quantitative Structure–Property Analysis of Boron α‐Icosahedral Nanosheet

In its crystalline state, the α‐icosahedral nanosheet of boron demonstrates superconductivity and thermal electronic properties. Mathematical research on a graph’s structure yields a graph descriptor, a numerical measure. Chemical graph theory employs connectivity descriptors to analyze molecular structures, providing crucial insights into many chemical compounds’ chemical and biological characteristics. These characteristics benefit physicists, chemists, and medical and pharmaceutical specialists. In this paper, the idea of reverse degree–based ‐polynomial is initiated, and differential and integral operators are computed. We formulate reverse degree–based topological descriptors based on this concept. In this paper, we examine the boron α‐icosahedral nanosheet for this technique. We looked at the physicochemical properties of boron α‐icosahedral nanosheets using reverse degree–based topological descriptors and best‐fit linear regression models to get an idea of what they are. Researchers are hoping that this strategy will lead them into new areas where they can investigate related studies.

Computation of Weighted PI Index of Lexicographic product graphs and for Silicates Networks

The study of chemical compounds’ molecular structures is one of the most cutting-edge uses of graph theory, along with computer science, nanochemistry, network design in electrical and electronic engineering, and the depiction of graphs in Google Maps. The degree and distance between vertices of a graph are the basis for examining topological indices. The formula for computing the Weighted Padmakar Ivan index (WPI) of a graph G is PIw(G) = P e∈E(G) [(dG(u) + dG(v)][|V (G)| − NG(e)].

Assessing the Logistics Efficiency of Baltic Region Seaports Through DEA-BCC and Spatial Analysis

Efficient logistics is a key factor in the competitiveness of seaports, especially in regions such as the Baltic Sea, where ports play important roles as hubs in the European Union’s Trans-European transport network (TEN-T). However, there are a lack of comprehensive studies focusing on the logistics efficiency of Baltic Sea ports, especially those integrating technical and technological factors. This study aimed to assess changes in the logistics efficiency of 15 major ports in the Baltic Sea region between 2019 and 2023, taking into account the technological and infrastructure-related elements that influence port performance. The model developed by the authors integrates the nearest neighbour method for cluster identification, data envelopment analysis using the Banker, Charnes, and Cooper (DEA-BCC) model to assess the overall technical, pure technical, and scale logistics efficiency, and spatial autocorrelation analysis to explore spatial interactions. For the DEA-BCC model, constraints were defined for each port based on inputs (number and length of berths) and outputs (cargo and container volumes for 2019–2023). The spatial autocorrelation analysis examined the relationships among the Baltic Sea ports, container volumes, and logistic efficiency values derived from the DEA model. Recognizing the sensitivity of the weight matrix in previous studies, this paper introduced an enhanced two-factor weighting matrix that incorporated geographical distance and the port connectivity index, calculated by the United Nations Conference on Trade and Development (UNCTAD). The statistical reliability of the results was validated using z-scores and p-values. The results showed that the overall technical efficiency of the ports analysed during the period considered was 47.2%, the pure technical efficiency was 61.0%, and the average scale efficiency was around 76%, indicating that diminishing returns to scale dominated. The spatial analysis showed a strong correlation between port connectivity and efficiency, indicating that well-connected ports, such as Gdańsk and Gdynia, had a higher efficiency. The findings make a significant contribution to the understanding of the logistics efficiency of Baltic Sea ports and highlights the importance of regional cooperation, infrastructure improvements, and better connectivity strategies to improve the overall efficiency of seaports in the region.

Neighborhood version of molecular descriptors for benzene type ring graphs

This study explores the utilization of topological graph invariants, or molecular descriptors, to mathematically model chemical compounds. These descriptors are vital in quantifying the physio-chemical characteristics of a compound, and are commonly represented as shapes such as polygons, bushes, and grapes. Specifically, this research focuses on computing selected fifth multiplicative first and second Zagreb indices, third and fourth multiplicative general fifth multiplicative Zagreb indices, and other degree-based topological indices for the benzene ring graph ([Formula: see text] and the simple bounded dual of benzene ring graph ([Formula: see text]. The findings of this study are then compared to demonstrate the impact of these molecular descriptors.

Computational modeling of triangular γ-graphyne using advanced topological methods

Modeling and computation of molecular descriptors play a crucial role in understanding the properties and characteristics of novel materials like [Formula: see text]-graphyne, the next-generation wonder material. Triangular [Formula: see text]-graphyne is an intriguing two-dimensional carbon-based material with diverse potential applications due to its unique electronic, mechanical, and thermal properties. Triangular [Formula: see text]-graphyne, a distinctive form of carbon allotrope, has garnered substantial interest as a result of its extraordinary characteristics. The notable value lies in its ability to be utilized in diverse domains for potential applications. The interconnected mesh structure of Triangular [Formula: see text]-graphyne exhibits exceptional mechanical strength and exceptional electron-transporting properties. This makes it a promising material for the development of high-performance electronic devices, such as transistors and sensors. Furthermore, gamma graphyne possesses excellent thermal stability, which makes it suitable for applications in the field of energy storage and conversion. It has the potential to be used as a catalyst in fuel cells or as an electrode material in supercapacitors. In this study, we utilize graph theory to dissect the molecular structure of triangular [Formula: see text]-graphyne, leading to the derivation of specific mathematical formulas for essential degree-based molecular characteristics. These findings can be instrumental in examining the correlations between the structure and properties of [Formula: see text]-graphyne. This paper focuses on two structures made from hexagonal honeycomb graphite lattices, like triangular [Formula: see text]-graphyne and triangular [Formula: see text]-graphyne chains with respect to some degree-based topological indices. The results obtained will aid in the investigation of the structure-property relationships in [Formula: see text]-graphyne.

Subsidies and affordability: a social approach to water supply tariffs

ABSTRACT This study provides evidence of targeting the performance of subsidies and affordability of water supply services (WSS) in a low-income society marked by disparate socioeconomic indicators. We used sample microdata from the local census regarding income, as well as information about price, tariff structure, water consumption, distribution of consumer units by billed volume ranges, and the average cost of WSS by the service areas (SA) in the Federal District, Brazil, in 2019 and 2020/2021, periods in which there was a transition from the tariff methodology. The results showed average water consumption of 108 L/(inhabitant· day−1) and 399 L/(inhabitant· day1) in the lower and higher-income areas, respectively. Customers with a monthly consumption of less than 27 m3 were subsidized in both tariff cycles. The distribution of subsidies was regressive according to the connection and consumption index in the SA of lower income. Approximately a quarter of the families in the lowest income regions commit more than 3% of their income to consume water in the lifeline block of the WSS tariff structure. Findings also indicate that poorer households headed by women and children are more likely to have water poverty problems.

A quantitative structure–property relationship approach to predict the biological activities of newly discovered anti-cancer agents with applications in implications of drug development

There are multiple implications for topological indices (TIs) in the realm of anti-tumour, anti-cancer, anti-viral agents, anti-inflammatory, anti-microbial, anti-fungal, anti-oxidant, and enzyme inhibition potentials. One such utilisation involves predicting the effectiveness of compounds, where TIs shed light on these sulfonamides’ chemical structures and associated characteristics. Analysts can pinpoint the most suitable treatments for certain cancer types by examining the degree-based TIs of various compounds. This paper explores the functions of mathematical descriptors as TIs in estimating the anatomical and pharmacological features of emerging drugs utilised for chemotherapy for cancer. Analysing the compounds of sulfonamide is carried out utilising graph-based TIs computed by employing edge partitioning. The establishment of a QSPR model then follows using a linear regression approach. The conclusions highlight the significant capability of TIs as valuable assets in exploring and characterising compounds within the domains of anti-tumour, anti-cancer, anti-viral agents, anti-inflammatory, anti-microbial, anti-fungal, anti-oxidant, and enzyme inhibition.

Machine Learning Regression Models for Predicting Anti-Cancer Drug Properties: Insights from Topological Indices in QSPR Analysis

This study advances the prediction of anti-cancer drug properties by integrating machine learning regression techniques with topological indices derived from hydrogen-depleted molecular graphs. Focusing on distance-based metrics, we investigate how atomic spatial arrangements influence molecular characteristics, enhancing Quantitative Structure-Property Relationship (QSPR) frameworks. Our analysis reveals significant correlations between various topological indices and key molecular properties, including polarizability, molar refractivity, and boiling point. The regression models, particularly Linear and Ridge Regression, achieved high predictive performance, with R 2 values exceeding 0.80, low Root Mean Square Error (RMSE), significant F-test, and p-values below 0.05 for highly correlated properties. The importance of careful index selection is underscored, as models using indices with strong correlations show superior predictive accuracy. This approach offers a more robust framework for predicting physicochemical properties, enhancing the efficiency of screening processes and optimizing lead compounds in oncological research. By bridging theoretical modeling and practical drug discovery, this work has the potential to accelerate the development of more effective and targeted anti-cancer therapies.

SOME TOPOLOGICAL INDICES VALUES OF THE CONVEX POLYTOPE Dn AND ITS SUBDIVISION GRAPH S(Dn)

Graph theory has been studied different areas such as computer science, information, mathematics, natural sciences, and so on. Especially, it has been the most important mathematical analysis tools for the study the analysis of chemistry science. For example, the studies of descriptors in quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR) studies in the chemistry science have been used graph theory tecniques. Furtermore, topological indices have been also used to determine combinatorial properties of chemical graphs. In this paper, exact values of some eccentricity-based topological indices of the convex polytope Dn and its subdivision graph S(Dn) have been determined.