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

Abstract

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.