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 SinftySinfty, i.e., of the group of all (not just finitary) permutations of mathbbNmathbbN, is topologically isomorphic to the automorphism group of a countable graph.
(2) The abstract group of increasing homeomorphisms of mathbbRmathbbR, rmHomeo+(mathbbR)rmHomeo+(mathbbR), has no non-trivial actions on a set of size <2aleph0<2aleph0. So in particular, it cannot be represented as the automorphism group of a graph with less than continuum many vertices.
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