A HYBRID ANALYTICAL AND MACHINE LEARNING APPROACH FOR MODELING M-POLYNOMIAL BASED TOPOLOGICAL INDICES OF SILICATE AND OXIDE NETWORKS

Abstract

Topological indices serve as foundational numerical invariants of molecular graphs, mathematically encapsulating the topological architecture of chemical compounds to enable the high-fidelity prediction of their bioactivity and physicochemical attributes. While the analytical derivation of these indices via M-polynomials has been rigorously established in pure mathematics, mapping these invariants for hyperscale, complex crystalline networks remains a non-trivial computational bottleneck. This paper introduces an advanced hybrid methodology that synthesizes exact algebraic derivations with robust machine learning (ML) architectures. We mathematically model selected silicate (Chain Silicate, Silicate Sheet) and oxide (Cuprous Oxide, Bismuth Trioxide) networks, formulating exact closed-form expressions for their respective M-polynomials. Leveraging these polynomial formulations, we analytically extract fundamental topological descriptors, including the First and Second Zagreb indices, the Randić index, and the Harmonic index. Subsequently, a comprehensive, high-dimensional dataset is synthetically generated from these closed-form expressions to train and validate state-of-the-art predictive models (Linear Regression, Random Forest, and XGBoost). Comparative performance analysis demonstrates that tree-based ensemble learning models, particularly XGBoost, achieve near-perfect predictive accuracy (). This hybrid framework significantly accelerates the computation of topological descriptors for large-scale macromolecular networks, reducing algorithmic complexity to  inference time while strictly preserving analytical integrity. Ultimately, this work bridges the critical paradigm gap between algebraic graph theory and applied computational intelligence.