An automorphism of a graph G is defined as a permutation of its vertex set that preserves the edge structure, meaning that if {a,b} is an edge, then {p(a), p(b)} must also be an edge. This concept is closely related to isomorphisms, as an automorphism can be viewed as an isomorphism where the domain and codomain are the same graph. The discussion clarifies that while all automorphisms are isomorphisms, not all isomorphisms are automorphisms. Understanding this distinction is crucial for grasping the properties of graphs in mathematical contexts. The relationship between automorphisms and isomorphisms highlights the structural symmetries within graphs.
