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Understanding the topology of the state space has proven to be extremelyefficient for dynamical systems with a continuous state space. On the otherhand, for particle systems on finite simple graphs, it has not yet been subjectto deep investigation due to combinatorial hurdles and existing efficientspectral theoretic approaches to the analysis of classical particle systems. Inthe context of complex systems with heterogeneous interactions of particles,these techniques can break down due to intractability. This work provides atool box of results on the topology of state spaces of particles systems underthe sole conditions that exactly one particle moves at a time and no twoparticles may occupy the same vertex at the same time. The Johnson$\mathcal{J}(n,k)$ graph yields the overarching structure and we prove that anyparticle system with exclusion property and single particle movements may beembedded as a dynamical system on a sub-graph of $\mathcal{J}(n,k)$.

On sub-graphs of the Johnson $\mathcal{J}(n,k)$ graph induced by particle systems on simple graphs