Topological Indices Research Articles

Distance based topological characterization, graph energy prediction, and NMR patterns of benzene ring embedded in P-type surface in 2D network

Nanostructures are tiny objects at the molecular and microscopic scale, with carbon nanotubes being the most notable among them. The elements possess exceptional microelectronic properties and other unique characteristics. Researchers have recently focused on the mathematical features of these materials. Molecular descriptors are crucial in mathematical chemistry, particularly in QSAR and QSPR modeling. Topological indices hold a significant position among them. This study presents the precise formulation of the ten most crucial topological indices for a benzene ring positioned on a P-type surface within the highly symmetric 2D lattice BCZ48\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {BCZ}_{48}$$\\end{document}. We have incorporated the computed indices to develop a predictive model for the graph energy of the 2D lattice and, in addition, provided the NMR patterns and the HOMO-LUMO gap.

Loop analysis quantifying important species in a marine food web

Improving the predictive power of food web analysis is a major challenge. Identifying the relationships that link topological and dynamical features may help. We used the predictions of loop analysis about the effect of perturbations targeted to the components of Barents sea food web to quantify their sensitivity and community impact, that we summarized in two new indices, NI and NS. Using a multivariate analysis we interpreted the meaning of these indices in a benchmarking exercise using several well recognized indices of species topological (positional) importance. Our findings suggest that the information the two indices proposed here provides does not overlap with that of more diffused topological indices of positional importance (i.e. centrality indices). The former are express the dynamic consequences of the topology in which species are embedded, whereas for the latter such dynamical consequences are mostly hypothesized on a topological base. The indices of loop analysis are based on the effective role a species plays in passing the impacts to other species (NI) and their role as sinks of the perturbations entering anywhere in the system (NS). These two indices, in the end, reveal how the topology of the network affects the response of the species to perturbations and thus emphasize the interaction between topology and dynamics. Based on our results, the question related to conservation is whether to prioritize sensitive species, that can be more strongly influenced when others are perturbed, or species of high impact, that can more strongly influence the rest of the community if perturbed.

Some Reverse Topological Indices of Comet and Double Comet Graphs

Introduction: Chemical Graph Theory or CGT for short, is a transdisciplinary field of Mathematics wherein graphs are used to represent chemical compounds. Under this representation, the atoms of a chemical compound are expressed as vertices, while the bonds connecting these atoms are expressed as edges. Once the graph representation of a chemical compound has been determined, graph-theoretic techniques can now be used to determine various topological indices associated with the chemical compound. Topological indices are invariants of a graph that have predictive power for the chemical properties of a chemical compound. In CGT, trees, being connected and acyclic, have been an essential class of graphs. This is because, most compounds have acyclic molecular structures. In the 2023 study by Gowtham and Husin, various reverse topological indices of a family of trees called bistar graphs have been determined. Objectives: Motivated by the work of Gowtham and Husin, we determine some reverse topological indices of another family of trees called comets and double comets. Double comets are natural extension of bistar graphs. We also give a certain computational application of our results on double comet graphs. Finally, we investigate the relationship between the reverse topological indices of bistar graphs and double comets. Methods: Several important graph-theoretic concepts were used to analyze some important properties of comets and double comets, leading to the computation of their reverse vertex degree based topological indices. Results: The following reverse vertex degree-based topological indices for comets and double comets were determined: Reverse sum-connectivity, First reverse Zagreb, Second reverse Zagreb, Reverse arithmetic-geometric, Reverse geometric-arithmetic, Reverse Sombor, and Reverse Nirmala. A computational application of the results to a certain chemical compound (2,2,4,4-Tetramethylpentane) or C9H20 were also demonstrated. Conclusions: The results of the paper provided an extension to an existing result on the reverse vertex degree-based topological indices of bistar graphs by considering double comet graphs. Several topological reverse vertex degree-based topological indices were also computed for comet graphs. The computed topological indices were also applied in determining some invariants of a certain chemical compound. For future studies, we recommend that reverse vertex degree based topological indices of other family of trees will be considered.

Data-driven approximations of topological insulator systems

A data-driven approach to calculating tight-binding models for discrete coupled-mode systems is presented. In particular, spectral and topological data are used to build an appropriate discrete model that accurately replicates these properties. This work is motivated by topological insulator systems that are often described by tight-binding models. The problem is formulated as the minimization of an appropriate residual (objective) function. Given bulk spectral data and a topological index (e.g., winding number), an appropriate discrete model is obtained to arbitrary precision. A nonlinear least squares method is used to determine the coefficients. The effectiveness of the scheme is highlighted against a Schrödinger equation with a periodic potential that can be described by the Su–Schrieffer–Heeger model.

Exploring topological indices and entropy measure via rational curve fitting models for calcium hydroxide network

This work uses rational curve fitting models to investigate the complex link between topological indices and entropy metrics in calcium hydroxide (Ca(OH)2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Ca(OH)_{2})$$\\end{document} networks. Entropy measurements shed light on the complexity and disorder of these networks, whereas topological indices are essential instruments for comprehending the structural characteristics of chemical substances. Our goal is to find new patterns and connections by merging these two fields, improving materials science’s prediction capacity. We calculate many topological indices, such as the Randic, Balaban, and Zagreb indices, and examine the relationship between them and entropy measurements obtained from (Ca(OH)2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(Ca(OH)_{2})$$\\end{document} structural arrangement.

Determining the chemical ordering in nanoalloys by considering atomic coordination types.

The energetically most favorable chemical ordering of bimetallic nanoparticles can be characterized by combining global optimization algorithms and surrogate energy models. The latter approximate the energy of nanoalloys relying on structural descriptors, training models, and data. Here, we systematically evaluate the performance of highly data-efficient topological descriptors [Kozlov et al., Chem. Sci. 6, 3868 (2015)] for predicting the energies of metal nanoalloys with different chemical orderings. We also introduce a new descriptor based on atomic coordination types, which results in a less data-efficient and interpretable approach, but improves the general accuracy and the quantification of orderings in the inner parts of nanoparticles. The capacity of both the original and new approaches in combination with a basin hopping algorithm is illustrated by generating convex hulls of PdZn nanoalloys and predicting the resulting active surface site distribution as a function of particle composition. Finally, we show how these approaches can be combined with machine-learning adsorption models in electrocatalysis studies for a fast evaluation of the reactivity landscape of targeted nanoalloys.

Molecular Topology Index of a Zero Divisor Graph on a Ring of Integers Modulo Prime Power Order

In chemistry, graph theory has been widely utilized to address molecular problems, with numerous applications in graph theory and ring theory within this field. One of these applications involves topological indices that represent chemical structures with numerical values. Various types of topological indices exist, including the Wiener index, the first Zagreb index, and the hyper-Wiener index. In the context of this research, the values of the Wiener index, the first Zagreb index, and the hyper-Wiener index for zero-divisor graphs on the ring of integers modulo a prime power order will be explored through a literature review and conjecture.

Assessing the effect of torrent control structures on sediment continuity and connectivity

The study investigated the effect of torrent control structures on the sediment cascade in the Vegliato mountain basin (Italy) related to an intense rainfall event of 50 years return interval. The Index of Connectivity (IC) was exploited to analyze the interaction of structures with longitudinal sediment (dis)connectivity. Moreover, the Sediment Continuity Ratio (SCR) was used to assess the effect on sediment (dis)continuity. The SCR is a novel parameter considering for each torrent control structure, the net balance of sediment deposition and erosion, and the cumulative proportion of the sediment cascade arriving from upstream. The SCR emphasises which structure was more prone to continuity or discontinuity during an event and to what extent compared toother structures. Moreover, a multi-perspective framework was carried out to help the interpretation of the SCR results within the context of the study area. The results of the (dis)connectivity assessment showed that the torrent control structures impacted sediment dynamics by influencing the slope and flow confinement, which in turn affected the IC. The (dis)continuity assessment showed structures prone to continuity mainly located in the upstream part of the catchment, where most of the over 60000 m3of sediment was generated. In contrast, structures prone to discontinuity were located in the downstream part, where deposition processes were favoured during the analyzed period. A total of 65 % of the structures similarly affected both (dis)continuity and(dis)connectivity. Data on individual structure functions and maintenance conditions were also included, emphasizing the importance of these factors in planning mountain basins management interventions.The study proposed a new metric that, despite being based on a single large event and a single basin, still provides a useful approach to investigate the interaction between sediments and individual structures, the entire sediment cascade and the channel control system.

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

Sediment runoff formation in a small periglacial catchment on the King George (Vaterloo) island, Antarctica

Global climate change has most significantly affected the Polar Regions. The increase in air temperature has stimulated the melting of glaciers in the Arctic and Antarctic, which has contributed to changes in the formation of water and sediment runoff. However, there are very few quantitative estimates of the sediment redistribution in the periglacial catchments of the Polar Regions. Specific features of water and sediment runoff were studied within the catchment area of the Korabelnyj Stream located on the Fildes Peninsula in Antarctica near the Bellingshausen Ice Dome. The main aim of the study was to investigate the conditions for the formation of water and sediment runoff and to identify the proportional contribution of washout and erosion material coming from the periglacial and maritime parts of the catchment area to the sediment runoff of the stream. A set of methods and approaches, including: a) estimates of the sediment flow connectivity index; b) fingerprinting technique; c) hydrometeorological observations; d) large-scale geomorphological survey and others, was ap[1]plied to identify the conditions for the formation of surface runoff and washout, the mechanisms of sediment redistribution in various parts of the fluvial network and to quantify the proportional contribution of two main sediment sources to the sediment runoff of the stream. A fraction with a particle size of ≤63 μm was used for geochemical and spectrometric analyzes of soils and sediments. In total, the content of 34 elements was analyzed, i.e. 6 radioisotopes and 28 stable elements. It has been established that despite the significant differences in the relief of the near-glacial and maritime parts of the catchment area, the indices of sediment connectivity are quite close and amount to –1,79 and –1,35, respectively. A significant part of the material transported by temporary streams from the slopes of the catchment area is redeposited in relief depressions partially occupied by water bodies. The main volume of sediments, which is at least 60–66% of the total sediment runoff in the outlet section of the Korabelnyj Stream, comes from the periglacial part of the catchment area. This is due to the increased water discharge relative to the non-glacial part of the catchment area, which results from the melting of snow and ice accumulated on the ice dome, the high erosion of moraine deposits unprotected by vegetation, and the presence of an ice core in moraines, which prevents water filtration.

M, NM-polynomials Based on Reverse, Reduced Reverse Degree and Neighborhood Degree Based Topological Indices with Applications to Bond Energy of Y-Junction Nanotubes.

In graph theory, M polynomials like the matching polynomial are very crucial in examining the matching structures within graphs, while NM polynomials extends this to analyze non-matching edges. These polynomials are important in many fields, including chemistry and network architecture. They support the derivation of topological indices for protein structure analysis, network communication optimization, and drug design in QSAR/QSPR investigations. The aim of this paper is to define novel M and NM polynomials for different topological indices and to derive their closed-form expressions, specifically for Y-junction nanotubes. These new polynomials and indices are employed to create a robust QSPR model to predict bond energy in Y-junction nanotubes, that provide high accuracy and reliability in the model's statistical performance. This paper introduces new forms of M and NM polynomials tailored to specific topological indices related to reverse and neighborhood reverse properties. We derive closed-form expressions for these indices in Y-junction nanotubes. Furthermore, we develop a QSPR model to predict bond energy in Y-junction nanotubes using the newly defined indices. We define novel M and NM polynomials for various topological indices and derive precise expressions for Y-junction nanotubes. Utilizing these indices, we construct a highly accurate QSPR model (R² = 0.999) for predicting bond energy in Y-junction nanotubes, confirming the validity of our polynomial definitions and indices. We have presented new M and NM polynomials for different topological indices and derive their expressions specifically for Y-junction nanotubes. With these newly defined indices, we have developed a highly precise QSPR model to predict bond energy, achieving an R² value of 0.999. This work underscores the effectiveness of our polynomial definitions and indices in predicting material properties.

Topological Characterization of Coronoid Structures via Valency Based Topological Indices.

The number of atoms producing a bond with an atom a in a molecular structure (graph) is regarded as the valency of a. A number of molecular structures have been topologically characterized via a large number of valency-based topological indices, which correlate certain physio-chemical properties like boiling point, stability, strain energy, and many more chemical compounds. Our intention is to perform the topological characterization of primitive coronoid structures initially on the base of seven different valency-based topological indices. Fundamentally, our purpose is to provide the universal formula by using the topological characterization of any primitive coronoid structures with any valency-based topological index. The technique of atom-bonds partitions according to the valency of atoms besides counting principals are used to find the results. Exact values of valency-based topological index, namely the Randic index, sum connectivity index, harmonic index, atom bond connectivity index, geometric arithmetic index, forgotten index, and symmetric division index are calculated coronoid structures initially constructed with at most three benzene lyres. All these indices are valency-dependent structural invariants that have been used to elaborate several chemical properties of compounds of the structure. Moreover, we provide the general formula to compute any valency-based topological index for any type of coronoid structure constructed with any number of benzene lyres.

Novel topological reverse indices and entropies of armchair versus zigzag polyhex carbon nanotubes with spectroscopic applications

The study of nanomaterials with different sizes and shapes is of considerable recent interest. Polyhex carbon nanotubes form an important class of nanomaterials that find several applications. Topological properties and entropies of two phases of carbon nanotubes, namely the zigzag and armchair configurations, have been juxtaposed through the reverse degree-based topological indices. The entropies and topologies of the two phases of the carbon nanotubes are also computed and compared which reveal that the zigzag nanotubes exhibit greater entropies compared to the armchair nanotubes. Applications of the developed techniques to various spectroscopies including NMR and ESR spectroscopies, are also pointed out. In future studies, detailed analysis and applications of reverse, reduced reverse topological indices and entropy of various other complex chemical structures will be considered.

Estimating sediment delivery ratio using the RUSLE-IC-SDR approach at a complex landscape: A case study at the Lower Snowy River area, Australia

Understanding the dynamics of sediment transport and deposition in natural landscapes is critical to developing cost-effective mitigation measures to control soil erosion and protect ecosystems. However, none of a single existing model can quantify sediment delivery ratio (SDR) and the impact factors such as vegetation and geomorphology, especially in a complex landscape. In this case study, we applied an integrated approach including the revised universal soil loss equation (RUSLE) and the index of connectivity (IC) to assess hillslope erosion and SDR, namely RUSLE-IC-SDR, across a complex landscape in the Lower Snowy River area, Australia. The RUSLE factors were derived from a high-resolution (2 m) digital elevation model (DEM), digital soil maps, high-resolution rainfall data and remotely sensed fractional vegetation cover. A seven-class landform classification was delineated from the high-resolution DEM using a fuzzy logic landform model (FLAG). We further examined the impacts of rainfall, vegetation cover and geomorphology on sediment dynamics and distribution across the study area. Field and laboratory data from 10 plot sites across the study area were collected and used for model validation. This case study showed that the RUSLE-IC-SDR approach can assess the overall sediment budget and the impacts of rainfall, vegetation cover and geomorphology across a complex landscape. Findings from this study can identify and track the areas likely to generate high sediment yield for developing ecological restoration, feral animal management and other catchment management measures.

Topology of quasi divisor graphs associated with non-associative algebra

The visualization of graphs representing algebraic structures has increasingly gained traction in chemical engineering research, emerging as a significant scientific challenge in contemporary studies. Due to their practical applications, graphs associated with groups and various algebraic systems have garnered significant interest in academic literature. Chemical graph theory employs graphs to illustrate molecular structures, where vertices symbolize atoms and edges represent bonds. The structure of quasi divisor graphs linked to non-associative algebra, including quasigroups and loops, presents numerous potential implications and wider applications. These applications span multiple disciplines, such as algebra, combinatorics, computer science, and network theory. Latin squares are tabular representations of quasigroups, generalizations of group structure. Numerous classifications of quasigroups exist in academic literature, such as Moufang quasigroups, Bol quasigroups, and alternative quasigroups. However, the structural characteristics of quasigroups exhibiting the weak inverse property closely resemble those of groups. The overall purpose of this study is to see the interrelationship between non-isomorphic quasi divisor graphs and a particular class of G-loops. To achieve our results, we employ a variety of methods, including a combinatorial approach, degree counting techniques, vertex and edge partitioning methods, as well as graph-theoretical tools and analytical procedures. In this paper, we compute a few topological indices based on degree, distance, and degree-distance using the structural aspects of quasi divisor graphs of orders 2φ,22ϕ,23ϕ and 4ϕ1ϕ2 where φ is a positive integer and ϕ1,ϕ2 are odd primes with ϕ1<ϕ2. This work also includes polynomial forms and their geometrical interpretations connected to linear, quadratic and higher degree topological indices of quasi divisor graphs associated with a class of non-commutative weak inverse property quasigroups.

Bicubic regression analysis a novel approach for estimation of physicochemical properties of skin cancer drugs through degree based entropy

Skin cancer poses a significant risk to the healthcare system worldwide and is projected to increase substantially over the next two decades, particularly if not detected in its early stages. The primary aim of this study is to construct a quantitative structure-property relationship (QSPR) by correlating calculated entropies with topological indices and specific physical-chemical properties of pharmaceuticals, in order to enhance their usefulness. The bicubic regression model is constructed through degree-based topological entropies to perform the QSPR analysis for the prediction of physiochemical properties like polar surface area, complexity, molar refractivity, boiling point, and polarity of skin cancer drugs. It is also examined that degree-based entropies provide best-fit models for skin cancer physiochemical properties. We inspected five physio-chemical properties of these anticancer drugs and found this methodology suitable for predicting best-fit approximations regarding R2 value. It is expected that this approach will be very favourable to examining problems related to mathematical chemistry.

Molecular descriptors of symmetrically configured carbon nanocones via quotient graph technique

Carbon nanocones, characterized by planar networks primarily consisting of hexagonal carbon faces and a few pentagons in the core, hold significant promise in various fields such as electronics, medicine, and material research due to their exceptional properties. This study exclusively investigates the topological descriptors of symmetrically configured carbon nanocones, specifically nanocones with two and three pentagonal core respectively. By modeling these structures as chemical graphs and employing molecular descriptors, we aim to provide a quantitative analysis of the structures without laboratory experimentation. The quotient graph approach is employed to determine the distance based descriptors, namely wiener, szeged and Padmakar-Ivan indices for these symmetrically configured carbon nanocones. The analytically closed formulas determined could be useful in quantitative structure-property-activity relationship studies, which helps in predicting the physico-chemical properties of the underlying structures. Thus the findings in this work not only deepen our understanding of symmetric carbon nanocones but also pave the way for significant advancements in nanotechnology and materials science.

Impact Of Road Connectivity On Daily Livelihood Sustainability Of Commuters In Sandeshkhali-I CD Block, North 24 Parganas, West Bengal, India

Background: The study area has different infrastructural deficiencies with compare to other regions. The natural constraints for communication, Poverty, Infrastructural deficiency, Rurality, and Transportation barriers are the main problems in this region. This study has tried to determine the different road connectivity and their impacts on daily sustainable livelihoods among the commuters of Sandeshkhali-I CD block in North 24 Parganas district of West Bengal, India. Materials and Methods: The study has used the Primary and Secondary data to fulfill the objectives. To execute the objectives, road networks are differentiated to measure the road connectivity indices (Alpha, Beta and Gamma Index) from different Gram Panchayats (GPs) to find out the road network condition in this rural deltaic region of the Indian Sundarban. Using the Random sampling to collect the primary data through questionnaire from commuters at different busiest places in the study area. Based on different selected livelihood components, the daily livelihood sustainability index has been evaluated. To find out the relationship the road connectivity Indices (R.C.I.) and daily livelihood sustainability index (D.L.S.I.) are correlated with the help of Karl Person ‘r’ value. Results: The highest values of Alpha, Beta and Gamma Indices are 0.34, 1.89 and 0.65 with respect to Metalled and Unmetalled roads. The daily livelihood sustainability index value has been found 54.33 and 17.54 as the highest and lowest respectively. The positive and negative values are present in the correlation between R.C.I. and D.L.S.I. Conclusion: From the outcome, most of the GPs are under the low and very low livelihood sustainability category whereas, the rest of the GPs have medium to high livelihood sustainability status as per selected class. The study concluded that the development needs to be accounted for because of intra-regional imbalance among the road connectivity as well as daily livelihood sustainability that has already existed in this area

Topological coindices and QSPR analysis for some potential drugs used in lung cancer treatment via CoM and CoNM-polynomials

Topological indices are used to convert a chemical structure into a real number, usually to study the physicochemical and biological properties of molecules. The groundwork is prepared for the interpretation of the obtained data by processing with Quantitative Structure Property/Activity Relationship (QSPR/QSAR). In this study, the drugs lorlatinib, gefitinib, sotorasib, pralsetinib, crizotinib, adagrasib, alectinib, brigatinib, dacomitinib and entrectinib, which are potential to be used in the treatment of lung cancer, are discussed. Topological coindices are calculated with the help of CoM and CoNM polynomials obtained with the graph structures of these drugs. The relationship between topological coindices and physicochemical properties such as evaporation enthalpy, flash point, molar refraction, polarisation, surface tension, molar volume are investigated by QSPR analysis. At this stage, linear, logarithmic and quadratic regression methods have been used. The results show that the values of these topological indices are highly correlated with certain physicochemical properties of the used some drugs in the treatment of lung cancer. In addition, using comparative analysis, the actual values and the values calculated with the help of topological indices have been examined in terms of predictive ability. The findings of this search demonstrate topological indices’ potential as tools for cancer drug discovery and design.

The Landscape Ecological Quality of Two Different Farm Management Models: Polyculture Agroforestry vs. Conventional

Low-intensity, diversified agricultural land use is needed to counteract the current decline in agrobiodiversity. Landscape ecology tools can support agrobiodiversity assessment efforts by investigating biodiversity-related ecological functions (pattern–process paradigm). In this study, we test a toolkit of landscape ecology analyses to compare different farm management models: polyculture agroforestry (POLY) vs. conventional monoculture crop management (CV). Farm-scale analyses are applied on temperate alluvial sites (Po Plain, Northern Italy), as part of a broader multi-scale analytical approach. We analyze the landscape ecological quality through landscape matrix composition, patch shape complexity, diversity, metastability, and connectivity indices. We assess farm differences through multivariate analyses and t-tests and test a farm classification tool, namely, a scoring system based on the relative contributions of POLY farms, considering their deviation from a local CV baseline. The results showed a separate ecological behavior of the two models. The POLY model showed better performance, with significant positive contributions to the forest and semi-natural component equipment and diversity; agricultural component diversity, metastability; total farm diversity, metastability, connectivity, and circuitry. A reference matrix for the ecological interpretation of the results is provided. Farm classification provides a quick synthesis of such contributions, facilitating farm comparisons. The methodology has a low cost and quickly provides information on ongoing ecological processes resulting from specific farm management practices; it is intended to complement field-scale assessments and could help to meet the need for a partially outcome-based assessment of good farm practice.