Explore a conference talk from TQC 2023 that presents a groundbreaking framework for understanding quantum spin systems through graph theory. Delve into how quantum spin systems can be precisely described using noninteracting fermions when their frustration graph meets specific conditions - being claw-free and containing a simplicial clique. Learn how this unified approach extends both the Jordan-Wigner transformation and Fendley's four-fermion model solution, bridging local and nonlocal solutions in quantum systems of any spatial dimension. Discover the identification of cycle symmetries in claw-free frustration graphs and understand how these commuting symmetries enable the application of Fendley's solution method across symmetric subspaces. Examine the physical interpretation of fermion modes through operators and their connection to operator Krylov subspaces, while understanding how this framework strengthens the relationship between many-body physics and claw-free graph theory.