Is every adjacency preserving bijection a graph automorphism?

Question

Let $f:G \to G$ be a bijective function on a finite simple connected graph which preserves adjacency (if $x$ is adjacent to $x'$ then $f(x)$ is adjacent to $f(x')$). Is $f$ automatically a graph automorphism?

Answer 1

Yes, if you mean "$x$ is adjacent to $x'$ if and only if $f(x)$ is adjacent to $f(x')$. Of course for a finite graph there is no problem. A graph has no other structure than its vertices and edges, and your $f$ preserves those.

Answer 2

Yes.

Let $V$ be the vertex set and $E \subset V^2$ the edge set. Define $g : V^2 \to V^2$ by $g(x,y) = (f(x), f(y))$. Then $g$ is also a bijection (actually we only need that it is an injection). The given assumption is that $g$ maps $E$ into itself; since $g$ is an injection and $E$ is finite, $g$ maps $E$ onto itself. So if $f(x) \sim f(y)$, we have $g(x,y) \in E$; since $g$ maps $E$ onto $E$, there exists $(u,v) \in E$ with $g(u,v) = g(x,y)$. But $g$ is injective so $(u,v) = (x,y)$. Thus $(x,y) \in E$, i.e. $x \sim y$. That completes the proof that $f$ is a graph isomorphism.

Connectedness is not needed here, but of course finiteness is.