Edge automorphism group

Sabidussi

Let $X$ be a simple, undirected graph. The vertex automorphism group, $\mathrm{Aut}(X)$, is isomorphic to the edge automorphism group, ${\mathrm{Aut}}_E(X)\cong \mathrm{Aut}(X{)}^E$, if and only if $X$ has at most one isolated vertex and $K_2$ is not a connected component of $X$.