It is everywhere, but I cannot see it from scratch. For example the graph $K_{1,3}$ given here (p23), it is easy to see why it is edge-transitive and not vertex-transitive. The definition given in Algebraic Graph Theory (Godsil) is:
A graph $G$ is edge transitive if its automorphism group acts transitively on $E(G)$.
The Petersen graph has $S_5$ as its automorphism group. The brute force approach will be to prove that all $S_5$ acts transitively on the edge set, but it is very tedious, right? Is there an easy way to see the edge transitivity?