How is Aut(G) = Aut(G¯)? (where G¯ is the complement of G)

Question

For a Graph G, I am trying to understand how the automorphism of G is equivalent to the automorphism of the complement of G.

I understand that an automorphism is an isomorphism of G onto itself. This, in fact, seems like it should be a fairly basic concept and so my confusion is likely in relation to what abstract concept an automorphism actually manifests as, and therefore how the equivalency can be seen.

Answer 1

Because $$(f(i),f(j))\in E(G) \iff (i,j)\in E(G)$$ is equivalent to $$(f(i),f(j))\not\in E(G) \iff (i,j)\not\in E(G),$$ which is equivalent to $$(f(i),f(j))\in E(\overline{G}) \iff (i,j)\in E(\overline{G}).$$