See Graph theory

A (graph) automorphism is an isomorphism from a graph to itself, i.e., where $G=G'$.

Automorphisms capture the notion of symmetry for a graph because imposing the above condition of edge preserving is that same than imposing that if we move vertices in a geometrical representation of a graph from their positions to the positions previously occupied by other nodes while carrying their connections with them (because if connections exist between $i$ and $j$, they must exist between $f(i)$ and $f(j)$, so that it is a homomorphism, i.e. a structure preserving map), then the new connections will be the same as those of the original graph (because the homomorphism property implies they are a subset of the connections. However, for it to be an isomorphism, the inverse map must also be a homomorphism, so that a connection $(k,l)$ must correspond to connections $(f^{-1}(k), f^{-1}(l))$, so that it is also a superset, and so the sets of edges are equal).

->Another way of looking at a graph automorphism is as a permutation $\lambda$ of the node labels $V$, such that a pair of vertices $(i,j)$ are connected if and only if $(\lambda(i), \lambda(j)$ are connected.

->Yet another way of looking at graph automorphisms is, I think, as symmetries of the Adjacency matrix. Any permutation of the node labels that leaves the adjacency matrix unchanged is a graph automorphism.

The set of all automorphisms of an object forms a group, called the automorphism group. Intuitively, the size of the auto- morphism group A ( g ) provides a direct measure of the abundance of symmetries in a graph or network. Every graph has a trivial symmetry (the identity) that maps each vertex to itself.

http://mathworld.wolfram.com/GraphAutomorphism.html