TOPOLOGICAL INDICES OF MOLECULAR GRAPHS AND THEIR APPLICATIONS TO STRUCTURE-PROPERTY PREDICTION

Topological Indices of Chemical Graphs and Their Applications in QSPR and QSAR Modelling

M-polynomial Based Topological Analysis and Comparative Study of SiC3 Networks

Topological indices are numerical descriptors used in chemical graph theory to characterize the size, branching and connectivity of molecular structures. These descriptors are significantly correlated with a range of physicochemical characteristics and biological activities of a molecular compound. Recently, the M-polynomial approach has been used to represent molecular structures and calculate degree-based topological indices for various graph structures. This study focuses on determining closed-form expressions of the M-polynomial for three types of silicon-carbon structures: SiC3-I[a,b], SiC3-II[a,b] and SiC3-III[a,b], for arbitrary a>1, b≥1. Using the obtained M-polynomial, we calculate nine degree-dependent indices for these silicon-carbide structures. The study also includes visualizations of the computed topological indices and the M-polynomial. Furthermore, we have done a novel comparative analysis among the topological indices for these specific structures. The results obtained in this study provide a mathematical foundation for future researchers in the property prediction of these structures.

Degree-Based Topological Descriptors and Physicochemical Structure–Property Trends in Oligothiophene Derivatives

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.

Physics-Based Molecular Fingerprints from Spectral Graph Theory Provide Efficient Geometry-Aware Measures of Chemical Similarity

Molecular representations are essential for the evaluation of molecular similarity and the development of structure-property relationships. Despite the known importance of 3D structure to determine chemical and physical properties, the most widely used molecular fingerprints encode only two-dimensional connectivity. Such representations fail to distinguish similar but distinct stereoisomers and conformers. Alternative 3D methods are typically defined pairwise, making their application to large chemical spaces prohibitive, while deep learning embeddings are expressive but uninterpretable and limited by their training data diversity. Here, we introduce novel physics-inspired molecular fingerprints based on principles from spectral graph theory. We represent molecules as a complete graph in 3D space, with edge weights encoding heuristic physical interactions. Eigenvalue decomposition of the resulting graph Laplacian matrix results in a computationally efficient fixed-length chemical fingerprint that encodes 3D structure while obeying necessary physical symmetries of permutation and E(3) invariance. Spectral fingerprints differentiate between unique molecular structures with identical 2D connectivity, overcoming a limitation of 2D descriptors, while maintaining the low computational cost needed for efficient screening of vast chemical spaces. We evaluate our fingerprints with community detection algorithms and observe strong performance against representative baselines across datasets from organic, inorganic, biological, reticular, and reaction chemistry. Nearest-neighbor property estimation and applicability domain analyses reveal the utility of our molecular representation in machine learning and cheminformatics. We anticipate that spectral fingerprints will serve as generalizable, interpretable, and efficient measures of chemical similarity that incorporate 3D information at minimal cost.

Revan Degree-Based Topological Indices of Some Generalized Thorn Cog Graphs

Topological descriptors based on Revan vertex degrees provide an effective framework for analyzing the structural characteristics of complex graphs. This work investigates Revan degree-based topological indices for generalized thorn cog graphs, including the first and second Revan indices, the Atom-Bond Connectivity Revan index, and the Geometric-Arithmetic Revan index. Closed-form expressions are derived for thorn cog complete, wheel graph, and star graphs, along with their special cases, using an edge- partitioning approach based on Revan degrees. The results offer a systematic method for evaluating these indices and demonstrate their relevance in modeling physicochemical and biological properties of chemical compounds.

Exact Distance-Based Topological Indices Of Double Star Double Fanbell Graphs With Applications In Chemical Graph Theory, QSPR/QSAR Modeling, And Drug Discovery

The topological indices are fundamental tools in chemical graph theory with the aim of numerically describing the structural properties of molecular graphs and also predicting physicochemical and biological parameters. In this paper, we present a new graph family called Double Star Double Fanbell (BSDF) which consists of double star graph with each pendant vertex added with a double fan graph. Eight important distance based topological indices are considered, namely Wiener index, Hyper-Wiener index, Harary index, Reciprocal Complementary Wiener index, Wiener Polarity index, Terminal Wiener index, Reverse Wiener index, and Reciprocal Reverse Wiener index are considered and exact closed-form expressions are obtained. The analytical formulations are derived by a systematic distance-partitioning method and written in terms of the graph parameters, which allows the formula to be computed efficiently for any graph size without a need for any repeated shortest-path computations. In addition to their mathematical importance, the calculated topological descriptors are useful molecular descriptors in Chemical Graph Theory, in which atoms are depicted as vertices, and chemical bonds as edges. These descriptors can be used effectively in Quantitative Structure–Property Relationship (QSPR) and Quantitative Structure–Activity Relationship (QSAR) models to predict molecular stability, boiling point, melting point, solubility, lipophilicity, biological activity, toxicity and pharmacokinetic properties. Moreover, the proposed BSDF graph provides a proper model for the highly branched organic compounds, dendrimer, hyperbranched polymers, and functional nanomaterials. Expressions obtained in this work are in an exact form, which is very appealing for large scale molecular databases, virtual screening, cheminformatics and AI-assisted drug discovery. Therefore, the suggested graph is not only playing a theoretical role in advancement of graph theory but also in the present day computational chemistry and pharmaceutical research.