Exploration of Multivariate Polynomials through Independent Sets of Sierpinski Rhombus Type Fractal Graph and its Iterations

Fractals, with their intrinsic self-similarity and structural elegance, have become powerful tools for modeling complex graphical structures. Among these, self-similar graphs are pointed out for their recursive node connectivity and hierarchical organization governed by fractal iterative principles. This study focuses on the Sierpinski Rhombus graph, a quintessential example of fractal structures, and introduces a computational approach for exploring its multivariate independence polynomials across iterative levels n = 1, 2, 3. Through a systematic algorithm, we construct the multivariate independence polynomials through independent sets, their cardinalities, and their inverses using SAGE mathematical software, thereby unveiling deeper algebraic patterns embedded within the graph's structure. Beyond their combinatorial significance, these multivariate independence polynomial descriptors serve as meaningful and interpretable feature for machine learning models, and bridges the gap between structural complexity and data-driven modeling of graph network with intelligent predictive analysis. The synergy of fractal-based graph polynomials with machine learning algorithm in this work holds immense potential thereby positioning this formulated approach as a promising foundation for advancements in complexity-aware learning and predictive modeling of fractal graphs in graph-theoretic intelligence.