Independent set-based multivariate graph polynomials for fractal-type silicate triangle structures

Silicate structures are the mineral materials widely studied for their extraordinary structural complexity and versatility. They naturally exhibit self-similarity, making them well-suited for analysis through chemical graphs, graph polynomials, and fractal theory. Fractals, characterized by their self-replicating patterns and intricate structures, have found extensive applications across various domains, particularly in the study of graphical structures. Independence polynomials serve as mathematical representations of molecular and chemical graph structures and have proven to be a powerful tool for analyzing the intricate structural adjacency relationships in graph molecular structures. This article explores the fractal-type silicate triangle graph, characterized by recursive and iterative patterns, as a distinguished class of self-similar graphs. An innovative computational approach is presented for evaluating their independence polynomials and multivariate independence polynomials at specific iterations $n=\mathrm{1,2}$ and 3. This proposed algorithm systematically identifies independent sets and computes the corresponding single-variable, multivariable polynomials and its inverse by employing SAGE mathematical software. A graphical analysis is performed for specific iterations, providing deeper insights into the combinatorial and structural characteristics of these representative graphs.