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First some setup:

Definition 1.9 A graph $\Gamma$ consists of a set $V(\Gamma)$ of vertices and a set $E(\Gamma)$ of edges, each edge being associated to an unordered pair of vertices by a function "Ends": $\text{ENDS}(e) = \{v,w\}$ where $v,w \in V$. In this case we call $v$ and $w$ the ends of the edge $e$ and we also say $v$ and $w$ are adjacent.

We also allow the possibility that there are multiple edges with the same associated pair of vertices. Thus for two distinct edges $e$ and $e'$ it can be the case that $\text{ENDS}(e) = \text{ENDS}(e')$. We also allow loops, that is, edges whose associated vertices are the same.

Definition 1.14 A symmetry of a graph $\Gamma$ is a bijection $\alpha$ taking vertices to vertices and edges to edges such that if $Ends(e) = \{v,w\}$, then $Ends(\alpha(e)) = \{\alpha (v),\alpha(v)\}$. The symmetry group of $\Gamma$ is the collection of all its symmetries. We note this group by $Sym(\Gamma)$

I take it that a symmetry is a map from $V(\Gamma) \cup E(\Gamma) \to V(\Gamma) \cup E(\Gamma)$.

Let $\overline{f} : V \to V$ be some bijection. In any natural way, does $\overline{f}$ induce a symmetry on $\Gamma$? I.e., does there exists a symmetry $f$ such that $f \bigg|_{V} = \overline{f}$? Here's what I thought. If $v \in V$, simply define $f(v) = \overline{f}(v)$. Now for the edges. Let $e \in E$. Then $\text{ENDS}(e) =\{v,w\}$ for some $v,w \in V$. Define $f(e)$ such that $\text{ENDS}(f(e)) = \{f(v),f(w)\}$.

But this doesn't seem rigorous, particularly that last sentence; and I'm worried about the possibility of multiple edges (which of the multiple edges, if there are any, is it being sent to?). I think it's clear that it preserves adjacency, but how do I verify that it's bijective?

Injectivity: Let $x,y \in V \cup E$ be such that $f(x) =f(y)$. Obviously $x$ and $y$ can't belong to different sets. If $x,y \in V$, injectivity of $\overline{f}$ implies $x=y$. If $x,y \in E$, then $Ends(x) = \{v_x,w_x\}$ and $Ends(y) = \{v_y,w_y\}$, so $Ends(f(x)) = Ends(f(y))$ or $\{f(v_x),f(w_x)\} = \{f(v_y),f(w_y)\}$...?

Surjectivity: Let $x \in E$ (if $x \in V$, there's nothing to prove). Then there are $v,w \in V$ such that $Ends(x) = \{v,w\}$. By surjectivity of $\overline{f}$, there are $v',w' \in V$ such that $f(v') = \overline{f}(v') = v$ and $f(w') = \overline{f}(w') =w$...But the problem is that $v'$ and $w'$ might not be adjacent, right...?

The reason for my interest is that nearly every example in my book defines a map just on the vertices or just on the edges and simply asserts that it is a graph symmetry. My thought was that the author was presupposing this result I'm asking about.

EDIT

Ah, I think I came up with a counterexample: a square with a line segment connecting the bottom right vertex to a vertex not on the square (so the graph has five vertices). Define $f$ to permute the bottom left vertex and the vertex not on the square, fixing everything else. This defines a bijection on the vertices. But if the follow the above procedure, you get something that maps an edge to a non-edge.

I think the above procedure works for complete graphs. Any other graphs?

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The situation that $\overline f$ induces a symmetry on $\Gamma$ is by far the exception rather than the general pattern.

You can see this easily by drawing a random graph on a piece of paper. Here is the first example that I found on the web, in which you will see many examples of vertex pairs which are ends of edges, and other vertex pairs which are not; no graph isomorphism can take one kind of vertex pair to the other.

The trouble with your proof scheme is this: assuming your definition $f(v) = \overline f(v)$, and given $e \in E$ such that $\text{ENDS}(e) = \{v,w\}$, there is no expectation in general that there even exists an edge whose ends are $\{f(v),f(w)\}$.

It's easy to see in fact that every example can be obtained from some complete graph $G$ on $p$ vertices by the following alteration: choose one constant $q$ and replace every edge of $G$ between a distinct pair vertices by $q$ edges between those same vertices; choose another constant $r$ and insert $r$ loop edges at each vertex. The parameters $p,q,r$ completely determine such graphs up to isomorphism.

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