There is some "asymmetry" between the definition of automorphism for graph on the one hand and group, ring, field, etc., on the other. This is not necessary.
One could define a morphism of graphs (V, E) ->
(V', E') as a map f : V -> E
such that {f(x),f(y)}
E' whenever
{x,y} E.
Then an isomorphism of graphs is a bijective morphism whose inverse is also a morphism
(in contrast to the ring case, this requirement is necessary),
and an automorphism of the graph (V, E)
is an isomorphism
(V, E) ->
(V, E).
We stayed away from this approach as we do not use the notions any further.