Topological invariants are key tools for studying the physicochemical and thermodynamic properties of chemical compounds. Recently, a new bond-additive distance-based graph invariant called the Mostar index has been developed. It measures the importance of individual edges and the graph as a whole. It is denoted and defined as MoG=& sum;xy is an element of EGnxxy-nyxy. This invariant is helpful to characterize the structure of a given connected graph G. In this invariant, the quantity nx(xy) means the number of vertices closer to x than y. In the present study, first, we have considered the Mostar index of extremal unicyclic graphs of n vertices with given diameter. Second, we have determined all unicyclic graphs that contain the second maximum Mostar index.

An analytical framework Merging QSPR modeling and spectral graph theory is employed to explore the properties of chosen antidepressant drugs. Spectral descriptors that capture important graph invariants, such as global connectivity, hierarchical structure and topological complexity of molecular structures, were mapped from the eigenvalue spectrum of the adjacency matrix. These spectral descriptors were the basis on which a novel high-order nonlinear third order surface-fitting model is developed. The proposed approach is capable of intricate molecular interactions, which provide accurate predictions of major physicochemical characteristics of pharmaceutical research and development. The purpose of the paper is to establish a relationship between two adjacency spectrum based descriptor and the physicochemical property of the anti-depression drugs that is determined by Y-Randomization method, achieving the strongest correlation coefficient (r=0.96) and co-efficient of determination (r^2=0.92) , indicating a very strong positive agreement for predicting the properties. The robustness of the proposed models for the each experimental data is assessed by Leave-One-Out Cross-Validation (LOOCV) which is moderate approximately. All the presented descriptor based models are significant, efficient and strong ability to model the training dataset as determined by statistical invariant; Pearson Correlation Coefficient r lies between 0.85 to 0.95, r^2 lies between 0.72 to 0.92, p-values lies between 10^-6 and 10^-12 , RMSE lies between 65 to 115, MAPE lies between 4 and 18. By visualizing the fitted surfaces, the complex relationship between molecular structure and physiochemical properties are investigated, offering valuable insights into how structural variations affect pharmacological properties and bolstering the predictive power of the suggested computational model. Future studies will involve generalisation of spectral descriptors of Laplacian and normalized Laplacian matrices, higher order surface fitting model integration, and spectral feature incorporation with machine learning models. The framework can potentially fast-track virtual drug screening, mathematically design-optimize molecular design, and offer a scalable and mathematically-based tool of next-generation computational drug discovery.

Magnesium oxide (MgO) is a prototypical ionic solid with a well-defined rock-salt lattice and is frequently used as a reference structure in materials science and theoretical modeling. In this work, we present a rigorous graph-theoretical characterization of an idealized MgO lattice, formulated as a periodic point-lattice for analytical purposes. The structure is treated as a two-dimensional topological projection of the three-dimensional rock-salt lattice, enabling exact symbolic analysis without incorporating surface relaxation, reconstruction, or energetic considerations. Using edge partitioning and degree-based methods, we derive closed-form analytical expressions for a class of irregularity topological indices (ITIs) associated with the resulting periodic graph. These indices quantify degree asymmetry and structural heterogeneity purely in a combinatorial sense and serve as mathematical descriptors of connectivity patterns in finite lattice graphs. The primary contribution of this study is the exact derivation of these irregularity indices for an ideal rock-salt lattice, which, to the best of our knowledge, has not been previously reported. It is emphasized that the present work is formulated entirely within a graph-theoretical framework and does not establish direct correlations with experimentally measured physical properties. The observed scaling behavior of the indices reflects topological boundary effects inherent to finite periodic graphs and should not be interpreted as physical disorder in real crystalline materials. Instead, the results provide a foundational mathematical framework for future studies on non-ideal lattices, such as defective, doped, or reconstructed systems, where irregularity-based descriptors may be meaningfully related to material properties.

The research article reports a simple strategy through Vilsmeier Haack formylation for the synthesis of ethyl 1-formyl-1,2,3,6-tetrahydro-4-methyl-2-oxo/thioxo-6-phenylpyrimidine-5-carboxylates. The most facile position for the formylation in these dihydro pyrimidines (DHPMs) is 1-N position. Different spectral analysis was used to confirm the proposed structures of newly synthesized compounds. A broad range of human tumor cell lines; viz. Lung, Colon, Breast, Ovary, Prostate, Melanoma, CNS etc. were used to screen the synthesized compounds. POM computational studies were performed to discuss the atomic charges, molecular geometry, as well as the drug score analyses. It is found that the introduction of formyl group at N-1 position is not much beneficial for antitumor applications, in contrast to the introduction of –COOC2H5 residue at C-5 in pyrimidine, while cytostatic nature of the compounds is retained. The coexistence of two combined pharmacophore sites is indicated by the bioinformatic analyses. The cytostatic nature of all the substituted N-formyl derivatives is indicated by the values of IG50 > 100 µM. Based on the obtained cytostatic results; cytotoxicity of the pyrimidine core can be targeted in the future work, by hindering antiviral pharmacophore site with metal complexes formation.

Chemical Graph Theory provides a robust computational framework for analyzing molecular structures through graph-based descriptors and topological indices, offering valuable insights into their chemical behavior and therapeutic potential. This study explores the potential of novel graph-theoretical descriptors and Quantitative Structure–Property Relationship (QSPR) models to investigate the structural and physicochemical properties of prominent anti-osteoporosis drug molecules, including Alendronate, Bazedoxifene, Raloxifene, Ibandronate, and Zoledronic Acid. A comprehensive set of 21 topological indices based on degree and neighborhood degree were developed and tested for anti-osteoporosis drug molecules to provide a thorough structural analysis. The QSPR analysis identified strong correlations between these topological indices and key physicochemical properties. These results highlight the utility of theoretical and computer modeling in predicting drug properties, enabling chemists to streamline the design and optimization of novel anti-osteoporosis drugs. By combining advanced mathematical and computer modeling with molecular property prediction, this study underscores the potential of computational approaches to improve drug development and discovery for osteoporosis and related conditions.

Let Delta(n )denotes the triangular ladder graph. The multiplicative degree Kirchhoff index of a graph G with m number of edges is Kf(& lowast;)(G) = 2m & sum;( n)(i-2 )1/lambda (i). The term lambda(i )are the normalized Laplacian (NL) spectrum. In this study we have calculated the Kf(& lowast;)(G) through the lambda(i) spectrums for Delta(n) network. To compute our main results, we have used the techniques of decomposition theorem (DT).

Diabetes mellitus is a persistent metabolic disorder marked by disrupted glucose homeostasis, ultimately giving rise to serious complications across multiple organ systems. Although therapeutic options have advanced considerably, existing treatments remain inadequate for fully controlling disease progression, simulating the necessity for innovative strategies in drug discovery and optimization. In contemporary quantitative structure-property relationship (QSPR) studies, topological indices that encode the structural and connectivity characteristics of molecules play a pivotal role. These indices facilitate the prediction of essential physicochemical properties that inform drug likeness and therapeutic potential. In the present work, degree-based, degree-sum-based, and reverse-degree-based topological indices are examined under a bond partitioning framework for a series of antidiabetic drug molecules. The QSPR analysis reveals strong correlations between these indices and key physicochemical attributes, confirming their predictive relevance. Using these descriptors, predictive models are constructed to estimate properties such as boiling point, complexity, heavy atom count, molecular weight, molar refractivity, polarizability, flash point, molar volume, and enthalpy of vaporization. To enhance predictive performance, advanced machine learning algorithms including random forest and XGBoost are employed. These ensemble methods effectively capture complex, nonlinear dependencies between molecular descriptors and physicochemical properties, yielding robust and interpretable models.

Cardiovascular diseases, such as acute myocardial infarction, present ongoing global health challenges, necessitating advanced computational tools for drug evaluation and optimization. This study proposes a novel hybrid framework that integrates fuzzy artificial neural networks (FANN) with quantitative structure-property relationship (QSPR) modeling to accurately predict key physicochemical properties of anti-cardiovascular drugs. By leveraging fuzzy logic, the model effectively captures uncertainty and imprecision inherent in chemical and biological data, while QSPR establishes rigorous quantitative links between molecular structure and properties such as density, boiling point, ACD LogP, and enthalpy of vaporization. A dataset comprising clinically relevant cardiovascular drugs, including antiplatelet agents, antiplatelet agents (clopidogrel and ticagrelor), beta-blockers (metoprolol and carvedilol), and thrombolytics (alteplase), was used to train and evaluate the models. FANN consistently outperformed conventional models, including Random Forest, with superior predictive accuracy as reflected in low error metrics (MSE: 28.83, MAE: 3.39, RMSE: 5.37) and a high value approaching 0.99. The results demonstrate FANN's robustness in modeling nonlinear structure property relationships and its potential utility for virtual screening and early-stage compound prioritization. This research contributes to computational cardiology by offering a reliable, interpretable, and cost-effective strategy for predicting physicochemical properties of cardiovascular drugs.

This study investigates the potential of topological co-indices and their polynomials as computational tools for predicting the physicochemical properties of antiviral compounds targeting the Ebola virus. For an in-depth Quantitative Structure-Property Relationship (QSPR) analysis, we develop and compute topological co-indices using CoM and CoNM polynomials derived from the molecular structures of antiviral drugs, including Galidesivir, Chloroquine, Favipiravir, Amodiaquine, Azithromycin, Brincidofovir, and Clomiphene. These indices were correlated with some important experimental physicochemical properties via linear and curvilinear regression techniques, and their predictive models were developed. The results reveal strong correlations between the topological indices and physicochemical properties, underscoring their utility in drug discovery and design. A comparative analysis of actual experimental values and those predicted by the topological indices shows their predictive accuracy. To sum up, this study highlights the potential of topological co-indices as innovative tools for speeding up the discovery and optimization of antiviral drugs, especially concerning Ebola virus disease. The findings contribute to the expanding research focused on using computational methods to tackle emerging infectious diseases.

The stability of lithium-ion batteries is heavily impacted by the uncontrolled ‎formation of lithium dendrites, which pose a significant safety risk. The chemical and ‎structural properties of small lithium clusters are critical to controlling this phenomenon ‎and improving electrode potential. The use of traditional techniques, including large-scale ‎experiments and DFT calculations, provides useful insights but is often hindered by high ‎costs and long timeframes. This study addresses this challenge by proposing a novel ‎approach using the CoM Polynomial, a new tool in chemical graph theory. We construct ‎molecular graphs for minimized lithium cluster structures from previous studies and utilize ‎the CoM Polynomial to generate codescriptors for each cluster. This approach extracts ‎significant structural information, enabling a more robust analysis of cluster properties. ‎Curvilinear regression analysis then identifies highly significant regression equations and ‎informative codescriptors, revealing strong relationships between these codescriptors and ‎the binding energy of the clusters. Furthermore, lithium metal holds tremendous promise as ‎an anode material for lithium-ion batteries but faces challenges like dendritic growth and ‎unstable solid electrolyte interphase formation. Researchers are exploring porous graphene ‎frameworks as a potential solution. In this vein, we investigated the molecular structure of ‎hexagonal and rectangular porous nano graphene using the CoM Polynomial to obtain ‎analytical expressions for their topological codescriptors. This study offers insights that ‎can help researchers improve the safety, efficiency, and sustainability of energy storage ‎devices by better understanding the properties of lithium clusters and porous nano ‎graphene.

The melamine-based TriCF structures, specifically the compounds with the general chemical formula C3H6N6, belong to the triazine compound family. These covalent organic frameworks represent a novel class of materials. They are recognized for their remarkable thermal stability, layered structural design, and promising applications as synthetic lubricants. This research provides a comprehensive analysis of the topological descriptors for TriCF structures, particularly emphasizing novel neighborhood irregularity descriptors. Through the examination of degree sums in end-vertex neighborhoods and the development of novel neighborhood irregularity topological descriptor models, this study offers an optimal approach for characterizing and interpreting the structural features of triazine-based covalent organic frameworks (TriCF). The proposed topological descriptors significantly enhance our ability to classify, compare, and predict the chemical and physical behavior of molecular structures within complex systems. Moreover, the advanced graph-theoretic approach employed in this study highlights its potential in material design, offering valuable tools for optimizing the properties of melamine-based compounds in industrial applications.

In this study, a comprehensive predictive topological modeling framework was employed to investigate the quantitative structure-property relationships (QSPR) of naturally occurring anticancer chalcones. A series of degree-based topological indices, including the first and second Zagreb indices, first and second redefined Zagreb indices, sum-connectivity index, symmetric division index, Randić index, and harmonic index, were computed to represent key molecular structural features. Statistical and regression analyses were performed to correlate these indices with experimentally reported physicochemical properties of chalcones. The study focuses on eight naturally derived anticancer Chalcones, namely, Isoliquiritigenin, Butein, Cardamonin, Sappanchalcone, Licochalcone A, Millepachine, Xanthohumol, and Curcumin-selected based on their structural diversity and well-documented pharmacological potential. Correlations between the computed indices and several physicochemical parameters, such as molecular complexity, molecular weight, molar refractivity, and polarizability, were examined. The study also includes a comprehensive statistical evaluation of the regression models to assess the predictive accuracy, robustness, and reliability of the proposed descriptors.

For a graph Q=(V,E){\mathbb{Q}}=\left({\mathbb{V}},{\mathbb{E}}), the transformation graph are defined as graphs with vertex set being V(Q)∪E(Q){\mathbb{V}}\left({\mathbb{Q}})\cup {\mathbb{E}}\left({\mathbb{Q}}) and edge set is described following certain conditions. In comparison with the structural descriptor of the original graph Q{\mathbb{Q}}, the topological descriptor of its transformation graphs displays distinct characteristics related to the structure. Thus, a compound’s transformation graph descriptors can be used to model a variety of structural features of the underlying molecular structure and initiate a structural analysis. In this work, the concept of transformation graphs is extended giving rise to a novel class of graphs, the (r,s)\left(r,s)-generalised transformation graphs, whose vertex set is union of rr copies of V(Q){\mathbb{V}}\left({\mathbb{Q}}), and ss copies of E(Q){\mathbb{E}}\left({\mathbb{Q}}), where r,s∈Nr,s\in N, and the edge set are defined under certain conditions. Furthermore, this class of graphs is analysed with the help of first Zagreb index. Mainly, there are eight transformation graphs based on the criteria for edge set, but under the concept of (r,s)\left(r,s)-generalised transformation graphs, infinite number of graphs can be described and analysed.

Kudriavite (CdBi2S4) is an important semiconductor material with promising applications in various technological domains. This paper focuses on the comprehensive topological study of CdBi2S4 using graph theoretical modeling techniques. More specifically, the crystal structure of the compound CdBi2S4 is modeled via edge partitioning techniques of graph theory by deriving mathematical closed form expressions for certain important irregularity topological indices. This strategy provides a quantitative approach to assessing the heterogeneity and structural complexity of these chemical compounds. The results highlight the significance of irregularity topological indices in predicting material stability, electronic distribution, and molecular interactions. The integration of graph-theoretic principles with material science fosters advancements in optoelectronics, thermoelectrics, photovoltaics, catalysis, and biomedicine, paving the way for the development of next-generation functional materials. The advanced topological analysis presented in this paper highlights the potential of graph-theoretic approaches in material construction, providing valuable tools for optimizing the properties of CdBi2S4 compounds in industrial applications.

The inverse sum indeg index of a given graph G is symbolized with ISI and defined by sum of the weights d u d v d u + d v entire links u v ∈ G . We denote d u (resp. d v ) the degree of a vertex u (resp. v) of G. In this paper, we obtained the sharp lower bound of tricyclic graphs with respect to the ISI index of order n ⩾ 6 with size n + 2 . By using atoms as vertices and chemical bonds as edges, graph theory permits the representation of molecular structures as mathematical entities called graphs. Based on the above concept, we formulated the ISI index of octane isomers (OI) and benzenoid hydrocarbons (BH) and compared the values of ISI index with various degree-based TI's via their correlations and chemical properties. The structural analysis of octane isomers is an important application of this research, as the ISI index delivers insights into stability designs across various isomeric forms.

Molecular graph theory provides a powerful mathematical framework for representing chemical structures, where atoms and bonds are modeled as vertices and edges of a graph. Topological indices, derived from these graphs, serve as numerical descriptors capturing the structural features of molecules. These indices are widely applied in Quantitative Structure-Property Relationship (QSPR) analysis to predict the physicochemical behavior of chemical compounds. In this study, we investigate a novel class of bioactive polyphenols-namely ferulic acid, syringic acid, p-hydroxybenzoic acid, benzoic acid, vanillic acid, and sinapic acid-well known for their antioxidant, anti-inflammatory, antibacterial, anticancer, and antiviral properties. Using several widely recognized degree-based topological indices, we construct molecular graph models of these polyphenols and establish linear regression models correlating the computed indices with essential physicochemical properties. Our QSPR analysis demonstrates strong predictive correlations, highlighting the potential of graph-theoretical descriptors in rational drug design and bioactivity prediction. The results validate the utility of topological indices as efficient computational tools in cheminformatics, offering valuable insights for future applications in pharmaceutical chemistry and material sciences.

Inorganic networks such as silicate and oxide structures are of significant interest in scientific and industrial domains due to their structural diversity and stability. In this study, we employ graph-theoretical techniques to analyze the structural properties of these networks using face degree topological indices and their reverse degree modifications. The relationship between these indices and graph energies is investigated to gain insights into molecular stability and reactivity. Predictive mathematical models are developed to estimate the graph energies of higher dimensional networks based on the computed indices. The results reveal strong correlations between face degree indices and graph energies, offering a robust framework for predicting the graph energy of silicate and oxide structures.

Zeolites are microporous crystalline aluminosilicate materials with diverse applications in adsorption, catalysis, and separation. PWN-type zeolite, with its cubic framework and 8-membered ring channels, is a structurally distinct class of three-dimensional frameworks. In this paper, we describe the PWN structure using chemical graph theory and derive hybrid degree-based topological indices along with their corresponding entropy measures. These indices are then employed to establish regression energy models that relate topological indices to various energy components of PWN, yielding strong correlations and demonstrating the predictive power of the proposed descriptors. The models effectively capture the structural complexity of zeolites and predict their physicochemical properties, thereby addressing a gap in the study of three-dimensional frameworks.

Lyme disease, caused by the bacterium Borrelia burgdorferi and transmitted through infected black-legged ticks, remains a significant health concern due to its potential for severe complications, including arthritis, neurological disorders, and cardiac issues. Early diagnosis and treatment are essential to prevent these outcomes. This study explores the predictive potential of reverse degree-based entropy indices for analyzing the molecular structures of therapeutic compounds used in Lyme disease treatment. While the use of topological indices for predicting physicochemical properties is well-established, our research uniquely integrates reverse entropy indices with a computational framework to refine the prediction process. We focus specifically on antibiotic drugs such as doxycycline, ceftriaxone, Doxy 100, cefotaxime, Ceftin, Cefuroxime, Erythromycin, EryPed, Erythrocin Lactobionate, Ofloxacin, Moxifloxacin, amoxicillin, and penicillin G potassium—commonly used to treat Lyme disease—and leverage a novel Maple-based algorithm for calculating reverse degree-based entropy indices. SPSS software was employed to assess correlations between these indices and critical physicochemical properties, such as molecular weight (MW), complexity (C), molar volume (MV), and XLog P. Unlike traditional experimental methods mandated by regulatory authorities for Chemistry, Manufacturing, and Controls (CMC) processes, our approach provides a supplementary predictive framework to streamline early-stage drug property estimation. The results reveal that first reverse Zagreb entropy effectively predicts molecular weight, reverse atom bond connectivity entropy effectively predicts complexity, reverse augmented Zagreb entropy effectively predicts molar volume and reverse geometric arithmetic entropy effectively predicts molecular XLog P. This study not only advances the computational methodology by employing novel combinations of entropy indices but also builds on existing work by focusing on a specific subset of Lyme disease drugs. While this framework offers a cost-effective preliminary tool for predicting physicochemical properties, it complements rather than replaces rigorous experimental validation required for regulatory reporting. These findings lay the groundwork for integrating computational and experimental methods, potentially accelerating drug development and enhancing therapeutic precision for Lyme disease.