We present a graph-theoretic approach to simplifying generic many-body Hamiltonians. Our primary result introduces a recursive twin-collapse algorithm, leveraging the identification and elimination of symmetric vertex pairs (twins), as well as line-graph modules, within the frustration graph of the Hamiltonian. This method systematically block diagonalizes Hamiltonians, significantly reducing complexity while preserving the energetic spectrum. Importantly, our approach expands the class of models that can be mapped to noninteracting fermionic Hamiltonians (free-fermion solutions), thereby broadening the applicability of classical solvability methods. Through numerical simulations on spin Hamiltonians arranged in periodic lattice configurations and Majorana Hamiltonians, we demonstrate that the twin collapse increases the identification of simplicial and claw-free graph structures, which characterize free-fermion solvability. Finally, we extend our framework by presenting a generalized discrete Stone–von Neumann theorem. This comprehensive framework provides insights into Hamiltonian simplification techniques, free-fermion solutions, and group-theoretical characterizations relevant for quantum chemistry, condensed matter physics, and quantum computation.

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• Quantum information theory
• Quantum simulation