# Determining Graph Automorphisms by Determining Ways of Permuting Edges

Here is an example from the book Groups, Graphs, and Trees (Ignore example 1.15; I accidently included it in my snippet of the pdf):

From my understanding, the symmetry group of a graph $$\Gamma$$ consists of all bijections $$f : V(\Gamma) \to V(\Gamma)$$ such that $$v,w$$ are adjacent vertices if and only if $$f(v),f(w)$$ are adjacent vertices. So, to construct the symmetry group $$Sym(\Gamma)$$, we just need to look at the ways which we can map vertices to vertices in such a way that adjacency is preserved.

However, in this example, the author is merely looking at the ways to permute the edges. Strictly speaking, elements in $$Sym(\Gamma)$$ don't act on edges. Is there another notion of symmetry ("edge" symmetry) which corresponds to the usual symmetry ("vertex" symmetry)? Correspond to in the sense that the "edge" symmetry group is isomorphic to the "vertex" symmetry group?

• So, as a function what is the domain/codomain of a graph automorphism $f$? It can't just be $V(\Gamma)$. Is it $V(\Gamma) \cup E(\Gamma)$, with the stipulation that $f(V(\Gamma)) \subseteq V(\Gamma)$ and $f(E(\Gamma)) \subseteq E(\Gamma)$? – user193319 Apr 23 at 10:00
• Yes, it is defined on $V \cup E$, such that $F(V) = V$, $F(E) = E$ and if $\langle x, y\rangle \in E$ then $f(\langle x, y\rangle) = \langle f(x), f(y)\rangle$. – mihaild Apr 23 at 11:12
• @mihaild I'm still a little confused. Why does it suffice to keep the vertices fixed and just look at the ways of rearranging the edges in order to determine $Sym(\Gamma)$? – user193319 Apr 23 at 21:06