Degree-based graph entropies quantify structural heterogeneity in molecular networks. However, a unified entropy treatmentof polythiophene and its bipolaron counterpart, together with a direct assessment of descriptor redundancy, is still lacking.This study therefore derives closed-form expressions for fourteen degree-based topological indices and their Shannon-typeentropies for polythiophene PLP and the bipolaron network BIPLP, and examines whether the resulting entropy measures provide distinct structural information. The descriptors considered are the Zagreb, hyper-Zagreb, forgotten, Yemen, redefined Zagreb, Randi´c, Sombor, Yemen–Sombor, general sum-connectivity, geometric–arithmetic, and atom–bond connectivity indices. The edge partitions are determined from the degrees of adjacent vertices and then used to obtain formulas for all admissible values of p. Numerical values and plots show how the indices and entropies vary with p. The entropy sequences are highly correlated over the examined ranges, indicating that several descriptors carry nearly the same information. The formulas give exact structural invariants for the two network families and help identify a smaller, non-redundant descriptor set for later structure–property studies.

This study presents a graph-theoretical and descriptive structure–property analysis of the linear \(\alpha\)-linked oligothiophenesbithiophene, terthiophene, quaterthiophene, sexithiophene, and octithiophene. From the common edge partition of theirhydrogen-suppressed molecular graphs, explicit closed forms are derived for fourteen degree-based descriptors: the first and second Zagreb, forgotten, Yemen, first and second hyper-Zagreb, three redefined Zagreb, Randi´c, Sombor, Yemen–Sombor, geometric–arithmetic, and atom–bond connectivity indices. Numerical values are obtained for all five molecules. Ten physicochemical endpoints reported in PubChem—polar surface area, molecular weight, complexity, XLogP3-AA, heavy-atom count, boiling point, enthalpy of vaporization, flash point, molar refractivity, and molar volume—are analyzed using descriptive linear regression. Every considered index is an affine function of oligomer length; therefore, the fourteen univariate regressions have identical fitted values and goodness-of-fit statistics for a fixed endpoint, although their slopes and intercepts differ. To avoid redundant reporting, the main text provides a compact endpoint-level summary and one representative index model, while all 140 descriptor–endpoint equations are supplied in the supplementary material. The models describe trends within this five-member homologous series and are not presented as externally validated predictive QSPR models.

INTRODUCTION: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. METHODS: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. RESULTS: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. CONCLUSION:The results can be used to optimize the properties of dendrimers for specific applications.

Numerous applications in chemistry are enabled by chemical graph theory, which is a branch of graph theory. Numerical quantities derived from the chemical graphs of a molecule, known as topological indices and co-indices, are used to model the chemical and physical properties of molecules in quantitative Structure-Property relationships (QSPR) and quantitative structure-activity relationships (QSAR) research. Fortunately, chemical-based experiments have found a strong connection between Topological descriptors (topological indices and co-indices) of molecular structures and their Physicochemical Properties, such as boiling point, and toxicity of drugs. Although several research reports have contributed to the computation of topological indices of the benzenoid circumcoronene series, studies on the calculation of topological co-indices are limited. This paper focuses on some topological co-indices. Several formulas of topological co-indices such as first Zagreb, second Zagreb, forgotten, and Yemen co-indices have been derived for the benzenoid circumcoronene series. In addition, the paper introduced new topological indices and their co-indices such as Gaza, Quds, and Palestine indices and co-indices and their mathematical formulas of the benzenoid circumcoronene series. Moreover, some algorithms have been built using Python programs to implement the mathematical formulas that are generally derived.

Machine learning (ML) techniques have been used to improve university admissions in Yemen, which faces major challenges. There is a bias against disadvantaged applicants. This bias comes from relying too much on traditional academic metrics. At the same time, weak technology leads to inefficiencies in handling many applications. These problems hurt the fairness and ability of current admission systems. To address this, we apply ML techniques. Various models, including logistic regression, support vector machines, k-nearest neighbors, random forest, and gradient boosting were evaluated. Their performance was evaluated based on accuracy and fairness. Among these, gradient boosting achieved the highest accuracy of 92% along with a notable 20% bias reduction measured by the Disparate Impact Ratio (DIR) and the Equal Opportunity Difference (EOD). Although all ensemble models demonstrated superior scalability and fairness compared to traditional methods. These models effectively processed datasets of over 15,000 student records while maintaining performance. Unlike complex deep learning models, the proposed models are easier to interpret. They show the clear importance of the feature. Entrance exam scores and high school GPA are the main predictors. Furthermore, the framework incorporates fairness-aware optimization, effectively reducing biases in admission decisions and enhancing suitability for socioeconomically diverse applicant pools. The findings illustrate that ML-driven approaches can revolutionize university admissions by delivering scalable, equitable, and resource-efficient solutions, which are particularly beneficial for institutions in constrained environments.

Graph theory has emerged as an influential tool for communication network design and analysis, especially for designing hybrid network topologies for local area networks (LANs). LAN topologies often face challenges related to scalability, data traffic optimization, and security. Designing reliable and efficient hybrid LAN structures remains a critical problem in communication networks. This paper addresses the issue by proposing the application of graph labeling techniques, particularly H-irregularity strength, as a mathematical framework to model and optimize hybrid network topologies. The results illustrate the way theoretical graph labeling and practical network technology interact, offering a novel solution to LAN design problems. This study adds to the expanding field of graph theory applications in communication networks by relating graph theoretical ideas to actual network topologies. With an emphasis on the irregularity strength of particular graph families, the role of graph labeling in optimizing these topologies is explored in this study. The labeling methods that are given offer valuable insights into improving communication efficiency, guaranteeing LAN scalability, and optimizing network architecture. The theoretical underpinnings of the application of graph theory to communication network modelling are strengthened by these discoveries. Labeling methods are introduced in this study to capture topological irregularities and labeling constraints through the use of specialized graphs, such as the Dutch Windmill and Corona product graphs as they both have special labeling characteristics that can be used to improve network performance. In order to secure data and prevent network failures, a three-unit organization structure with a shared administrator is used in network design and optimization. A model of hybrid ring topology of a local area network is considered in this paper and different models are presented which are originated from Dutch Windmill and corona product of graphs. The contribution of this paper is it includes results about a special version of irregularity strength in which the subgraphs used are Dutch windmill graphs and cycle graphs. The edge, vertex and total H-irregularity strength of Dutch Windmill graph and Corona product graph are calculated, offering fresh perspectives on the way they could represent hybrid LAN topologies. The irregularity strength metric is particularly useful even though it measures the imbalance in vertex degrees, which is essential for optimizing communication flow and load balancing within a network. Our theoretical findings illustrate how these labeling schemes can model network behavior, improve resource allocation, and trace data flow effectively. Although the study is primarily theoretical, it offers groundwork for practical network simulation and real-world implementation. Future work will focus on validating the models through simulations and assessing performance metrics such as latency, throughput, and fault recovery. A key limitation of the current study is the absence of empirical performance validation, which is identified as an important direction for further research.

This work investigates the special thermophysical properties of ternary nanoparticles and suggests the best possible compositions for them. This study explores heat and mass transfer in viscous fluid flows driven by rotating porous discs, a key process in energy and chemical industries. Ternary nanoparticles greatly enhance the thermal characteristics of base fluids, which makes them useful for use in solar collectors, electronics, nuclear reactors, and energy storage. This study's examination of the interfacial morphology effects that occur between water and Cu, AlO3, and TiO2 nanoparticles in a flow between two rotating porous discs is a significant new breakthrough. Also taken into account are the effects of magnetic fields and permeability. The shooting method in conjunction with the RK scheme is used to solve the governing nonlinear PDEs once they have been converted to ODEs through the proper transformations. Soret number, Smidth Number and Dufour Number are benchmarked with previous findings to assure correctness, and comparative data visualization is used to highlight the findings. Extending previous research, it evaluates fluid dynamics under slip conditions, influencing heat and mass transfer rates. Key findings include a 15

Topological measures of molecular graph structure according to degree yield valuable tools that facilitate understanding material’s characteristics through their structures involving physical, chemical, and biological properties. In this study, we have computed the Y-index and Y – polynomial for specific chemical structures, namely V-Phenylenic Nanotube and Nanotorus, as well as their molecular complement graphs. By analyzing the data from the computations using MATLAB, we were able to determine the connection between Y – index and Y – polynomial describing the V-Phenylenic Nanotube and Nanotorus. This elucidation provided insight into the physicochemical properties inherent to these compounds.

Machine learning is a vital tool in advancing drug development by accurately predicting the physical, chemical, and biological properties of various compounds. This study utilizes MATLAB program-based algorithms to calculate topological indices and machine learning algorithms to explore their ability to predict the physio-chemical properties of asthma drugs. By combining machine learning with topological indices, we can conduct faster and more precise analyses of drug structures. As we deepen our understanding of the relationship between molecular structure and performance, the integration of machine learning with QSPR research highlights the significant potential of computational strategies in pharmaceutical discovery. The use of machine learning algorithms such as random forest and extreme gradient boosting is essential in this process. These algorithms leverage labeled data to predict complex molecular processes, aiding in the discovery of new medication options and enhancing their properties. These methods enhance the accuracy of physical and chemical property predictions, streamline the drug discovery process, and efficiently evaluate large datasets through machine learning. Ultimately, these advancements facilitate the development of innovative and effective treatments.

Angina is a condition characterized by chest pain or discomfort due to insufficient blood flow to the heart muscle. Effective management focuses on reducing symptoms and preventing disease progression through lifestyle modifications, medications, and interventional procedures. Timely diagnosis and treatment are crucial for enhancing patient quality of life. Designing and developing experimental drugs is challenging and costly, which makes mathematical and computational methods essential for efficient drug discovery. In this article, we introduce a novel molecular descriptor based on a graph theory-driven degree partitioning technique, integrated into a quantitative structure-property relationships (QSPR) framework. Using quadratic regression, we determine the optimal predictors for four key properties boiling point, enthalpy of vaporization, flash point, and index of refraction for sixteen anti-angina drugs based on nine degree-based topological indices. Furthermore, by combining these descriptors with the multi-attribute decision-making additive ratio assessment technique, we achieve robust and reliable drug rankings. Our innovative integration of a new molecular descriptor with advanced statistical and decision-making methods not only improves predictive accuracy but also provides a novel and efficient approach for the development and optimization of angina drug therapies.

This work studies hub domination and total hub domination in both standard and fuzzy graphs. We determine exact values for principal graph families (paths, cycles, complete graphs, complete bipartite graphs, and wheels) and develop structural bounds that relate hub parameters to classical invariants. On the fuzzy side, we formalize hub domination with vertex and edge memberships, compute the fuzzy hub domination number for standard fuzzy graph classes, and derive degree- and structure-based bounds. We connect these results to applications in telecommunication and transportation networks, where minimizing hub cost aligns with the fuzzy hub domination objective. The findings clarify the mathematical structure of hub domination, situate it among core graph-theoretic measures, and provide implementable bounds and algorithms for design under uncertainty. These contributions offer tools for cost-aware hub placement and resilient connectivity in complex networks modeled by fuzzy graphs.

One of the most significant issues in wireless sensor networks (WSNs) is security, which must be addressed to keep WSNs safe from malicious attacks. An intrusion detection system (IDS) is essential in analyzing network traffic and detecting abnormal events. However, these IDSs suffer from several drawbacks that affect their effectiveness and flexibility in accuracy, so they must overcome these drawbacks to improve the performance of IDS. These drawbacks include difficulties in determining the appropriate dataset, the problem of feature selection, and the issue of the imbalanced dataset and choosing the appropriate algorithms for the classification process in WSN. In this paper, a model for an anomaly-based IDS in WSNs is proposed. This model applied mutual information (MI) for feature selection and the synthetic minority oversampling technique (SMOTE) for solving the imbalanced dataset problem. It used different machine learning (ML) algorithms, random forest (RF), decision tree (DT), support vector machine (SVM), and K-nearest neighbors (KNNs) to analyze network traffic and binary classification or multiclass classification. To implement and evaluate the performance of the proposed model, the standard dataset NSL-KDD is used. Python language is used to implement the proposed model in the Anaconda platform, and many evaluation metrics are also utilized to evaluate the performance of the proposed method. Experimental results show that the proposed model can detect intrusions using different ML algorithms with high accuracy. The results of the proposed model for different ML algorithms outperform the state-of-the-art algorithms, and the maximum enhancement reached 15% in the accuracy metric.

The Y-index and coindex are degree based molecular structure descriptors that have been shown to give a high degree of predictability compare to Zagreb indices and F-index and their coindices for some physicochemical properties of octane isomers. In this paper, we studied the Y - index and Y - coindex for certain important chemical structures like line graphs of the V C5C7[p, q] and HC5C7[p, q] nanotubes and their molecular complement graph. Moreover, we defined Y - polynomial of graph G and applied it on the line graphs of the V C5C7[p, q] and HC5C7[p, q] nanotubes. These explicit formulae can correlate the chemical structure of molecular graph of nanotube to information about their physical structure.

An unsteady two-dimensional magnetized Casson nanofluid flow model is constructed over a wedge under the effect of thermal radiation and chemical reaction. The multiple slip effects are also assumed near the surface of the wedge along with the convective boundary restrictions. This study investigates the application of soft computing techniques to address the challenges posed by the complexity of problem modeling and numerical methods. Traditional approaches incorporating various model factors may struggle to provide accurate solutions. To resolve this issue, Gaussian process regression (GPR) is employed to predict the solution of the proposed flow model. With the help of the numerical shooting method together with Runge–Kutta–Fehlberg fourth-fifth-order (RKF-45) reference data, the GPR model is trained. The numerical simulation illustrated that the Casson fluid parameter β and the unsteadiness parameter S strengthen the friction factor, and the heat transfer rate is enhanced as the radiation parameter Rd becomes larger. In addition, the Biot numbers Bi1 & Bi2 lead to strengthen nanoparticle temperature; an opposite behavior is noticed with the skin friction coefficient S˜fxRex0.5, heat transfer rate H˜tx Rex0.5, and nanoparticle transfer rate C˜txRex0.5. The GPR model with the exponential Kernel function provided better performance than other functions on both training and checking datasets to predict S˜fxRex0.5,H˜tx Rex0.5, and C˜txRex0.5. Statistical metrics including RMSE, MAE, MAPE, MSE, R2, and R are employed to check the accuracy and convergences of the predicted and numerical solutions obtained through GPR and RKF-45. It is observed that all three GPR models had an R2 value of higher than 0.9. The proposed study demonstrates the advantages of employing soft computing methods (GPR) to effectively analyse the behavior of complex flow models.

The first and second Hyper-Zagreb indices are the molecular descriptors that describe the structures of chemical compounds and they help us to predict certain physicochemical properties. In this paper, we studied the first and second Hyper-Zagreb indices for certain important chemical structures like line graphs of the VC5C7 [p, q] and 5 7HC C [p, q] nanotubes. Moreover, we defined second Hyper-Zagreb-polynomial of graph G and applied it on the line graphs of the 5 7VC C [p, q] and 5 7HC C [p, q] nanotubes. These explicit formulae can correlate the chemical structure of molecular graph of nanotube to information about their physical structure.

Degree-based topological descriptors of chemical structures may help to measure the physico-chemical and nano-properties, which are necessary for the growing industry. Hexaphenylbenzene is a chemical structure that plays a significant role in a wide variety of applications in organic electronic materials and liquid crystals, crystal engineering, and other polymeric materials, which are among the most active research areas of investigation in chemistry, biology, and drugs. In this study, in view of the mathematical formulation and analysis of structures, some topological descriptors of the chain molecular graph of hexaphenylbenzene L-n are studied. Moreover, using MATLAB the results are analyzed and the relationship between topological indices and polynomials in the chain of hexaphenylbenzene L-n, which portrays the physio-substance properties has been deduced. These numerical values correlate structural facts, chemical and biological activities, and physical characteristics. The results obtained could help us learn more about the properties of the chain of hexaphenylbenzene.

A key paradigm in contemporary research is the use of graphs to represent physical systems, molecular structures, or particularly metal frameworks. Graphs are increasingly widely used in a variety of fields, including the study of quantum and molecular systems, macromolecules and their interactions, socioeconomic and ecological systems, and technical and infra-structural systems. Understanding how these systems function, are robust, and are stable begins with structural characterization. The use of entropies and entropy-like measurements of graphs/structures of molecules/networks is crucial from both a mathematical and physical standpoint. Several entropy measures of graphs have been defined and studied extensively during the last few decades. The current paper is devoted to investigation of distance dependent entropy measures of Poly Propylene Imine (PPI) dendrimers and Zinc Porphyrin dendrimers. The analytical formulae of distance dependent entropy measures have been developed and their patterns have been presented through graphical tools. Mathematical chemistry offers valuable tools including molecular descriptors to predict molecular features in compounds. This research focuses on developing distance-based entropy topological descriptors for Poly Propylene Imine and Zinc Porphyrin Dendrimers. Regression analysis, employing statistical tools, is then applied to construct linear models for the chemical characteristics of these compounds. The generated model facilitates the prediction of attributes in subsequent generations, serving as an efficient alternative to time-consuming experimental research for obtaining numerical property values. image

In the field of graph theory, the exploration of connectivity patterns within various graph families is paramount. This study is dedicated to the examination of the neighbourhood degree-based topological index, a quantitative measure devised to elucidate the structural complexities inherent in diverse graph families. An initial overview of existing topological indices sets the stage for the introduction of the mathematical formulation and theoretical underpinnings of the neighbourhood degree-based index. Through meticulous analysis, the efficacy of this index in delineating unique connectivity patterns and structural characteristics across graph families is demonstrated. The utility of the neighbourhood degree-based index extends beyond theoretical graph theory, finding applicability in network science, chemistry, and social network analysis, thereby underscoring its interdisciplinary relevance. By offering a novel perspective on topological indices and their role in deciphering complex network structures, this research makes a significant contribution to the advancement of graph theory. The findings not only underscore the versatility of the neighbourhood degree-based topological index but also highlight its potential as a tool for understanding connectivity patterns in a wide array of contexts. This comprehensive analysis not only enriches the theoretical landscape of graph descriptors but also paves the way for practical applications in various scientific domains, illustrating the profound impact of graph theoretical studies on understanding the intricacies of networked systems.

This paper deals with some types of topological indices called valency-based indices or degree-Based Indices. Specifically, Multiplicative Forgotten, Multiplicative Yemen, modified Forgotten, modified Yemen, generalized modified first Zagreb, generalized modified sum connectivity, and generalized modified product connectivity indices of the benzenoid circumcoronene series are computed. Moreover, the formulas of polynomials for all these topological indices were derived.

A topological index is a branch of chemical graph theory that is vital to analyzing the physio-chemical characteristics of chemical compound structures divided into a degree-based molecular structure such as Zagreb indices, a distance-based molecular structure such as Wiener index, and a mixed such as Gutman index. In this paper, some definitions, results, and examples of Wiener polynomial and index for subdivision graph of friendship, bifriendship graphs, line subdivision graph of friendship, and bifriendship graphs were introduced. Moreover, we used the MATLAB program to calculate the Wiener polynomial and index of these graphs and refer to some applications.