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Infinite graphs have been used as a "discrete version" of topological spaces, for instance infinite Cayley graphs as a discretisation of homogeneous spaces). Gromov constructed homogeneous spaces out of limits of infinite Caley graphs to prove his theorem on homogeneous spaces. Homogeneous spaces and Cayley graphs share the property that you can transfer any point (vertex) to any other by an auto-homeomorphism (isomorphism).