gr.group theory - Realizing groups as automorphism groups of graphs.

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 $S_infty$, i.e., of the group of all (not just finitary) permutations of $mathbb N$, is topologically isomorphic to the automorphism group of a countable graph.

(2) The abstract group of increasing homeomorphisms of $mathbb R$, ${rm Homeo}_+(mathbb R)$, has no non-trivial actions on a set of size $<2^{aleph_0}$. So in particular, it cannot be represented as the automorphism group of a graph with less than continuum many vertices.