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][1] 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). 

[1]: https://en.wikipedia.org/wiki/Gromov%27s_compactness_theorem_(topology)