  In this paper, we show that Frucht's theorem holds in Borel setting. More
specifically, we prove that any standard Borel group can be realized as the
Borel automorphism group of a Borel graph. A slight modification of our
construction also yields the following result in topological setting: Any
Polish group can be realized as the homeomorphic automorphism group of a
$\mathbf{\Delta^0_2}$-graph on a Polish space.
