Why the Petersen graph is edge transitive 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? 
 A: Regard the Petersen graph as the Kneser graph $KG_{5,2}$: that is, the graph whose vertices are 2-element subsets of $\{1,2,3,4,5\}$, with an edge between any two vertices which are disjoint as subsets. (The easy way to see that this corresponds to the standard "star" picture for the Petersen graph is to put all the sets of the form $\{a,a+1\}$ in the outer pentagon, and all the sets of the form $\{a,a+2\}$ in the inner star, with all addition performed $\bmod 5$.)
Now, each edge of the Petersen graph corresponds to a pair of disjoint 2-element subsets of $\{1,2,3,4,5\}$. As $S_5$ acts transitively on all such pairs, the Petersen graph is edge-transitive.
A: Looking at the standard picture for the Petersen graph, it is clear that the automorphism group acts transtitively on the edges of  exterior pentagon, on the edges of the inner star, and on the edges of connecting the two. It is enough then to find an isomorphism mapping one of the outer edges to one of the edges in the star and one mapping it onto one of the middle edges.
So it all comes down to finding those two automorphisms.
Notice that this is the brute force approach —which is not that tedious.
A: You can obtain the Petersen graph from the dodecahedral graph (aka. the edge graph of the dodecahedron) by identifying antipodal vertices (vertices of maximal mutual distance).
One also say, that the dodecahedral graph double covers the Petersen graph.
The dodecahedron is famously known to be a regular polytope, i.e. its symmetry group acts transitively on its vertices, edges and faces. The Petersen graph inherits the edge-transitivity via the covering from the dodecahedral graph.
See also "hemi dodecahedron".
