In the topological setting or if you want to relate the size of the graph to the size of the group, there are two relevant results:
(1) Any closed subgroup of , i.e., of the group of all (not just finitary) permutations of , is topologically isomorphic to the automorphism group of a countable graph.
(2) The abstract group of increasing homeomorphisms of , , has no non-trivial actions on a set of size . So in particular, it cannot be represented as the automorphism group of a graph with less than continuum many vertices.
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