An edge-transitive graph is a graph such that any two edges are equivalent under some element of its automorphism
group. More precisely, a graph is edge-transitive if for all pairs of edges 
there exists an element 
of the edge
automorphism group 
such that 
(Holton and Sheehan 1993, p. 28). Informally
speaking, a graph is edge-transitive if every edge has the same local environment,
so that no edge can be distinguished from any other based on the vertices and edges
surrounding it.
By convention, the singleton graph and 2-path graph are considered edge-transitive (B. McKay, pers. comm., Mar. 22, 2007).
A graph may be tested to determine if it is edge-transitive in the Wolfram Language using EdgeTransitiveGraphQ[g].
A connected undirected graph is edge-transitive iff its line
graph is vertex-transitive. Note that
this statement does not hold in general for disconnected graphs, since for example
the line graph 
of the disjoint graph union of the cycle
graph 
and the claw graph 
is isomorphic to the graph disjoint union 
of two triangle graphs,
which is edge-transitive, while the original graph is clearly not.
Counting empty graphs as edge-transitive, the numbers of edge-transitive graphs on 
,
2, ... nodes are 1, 2, 4, 8, 12, 21, 27, 39, 50, 69, ... (OEIS A095352).
The numbers of connected edge-transitive graphs on 
, 2, ... nodes are 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, ...
(OEIS A095424; c.f. Conder 2017).
A graph that is both edge-transitive and vertex-transitive is called a symmetric graph (Holton and Sheehan 1993, pp. 209-210). A regular graph that is edge-transitive but not vertex-transitive is called a semisymmetric graph.
