Automorphism I don't understand

[S&S]

A permutation p of the vertex set of a graph G with the property that {a,b} is an edge if and only if {p(a), p(b)} is an dege, is called an automorphism of G. Is this right? this sounds isomorphism to me.

 

[AKG]

An automorphism is an isomorphism whose domain equals its codomain. So you know the general notion of an isomorphism f : G -> H. Well an isomorphism f : G -> G is called an automorphism of G.