Algorithm for Counting Independent Sets on Regular and Irregular Sheets of Graphene

The presence of topological defects in graphene, such as Stone-Wales configurations, significantly alters its electronic and mechanical properties. Traditional methods for counting independent sets in such structures face exponential complexity and irregular structures, limiting their applicability to nanoscale resolutions. This work introduces a novel algorithm based on Fibonacci recurrence rules, capable of handling cyclic defects through memoization and load propagation. Validated on hexagonal sets of up to 10⁴ nodes, our approach reduces computation time by 15 orders of magnitude compared to brute-force methods, enabling accurate modeling of defect-engineered graphene. This advancement connects materials science with algorithmic design, providing a tool to predict stable configurations in materials.