An edge-transitive graph, also called a line-symmetric graph by Harary and Palmer (1973, p. 263), is a finite simple 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 graph automorphism
group 
whose induced action on edges satisfies 
(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 finite connected simple graph with at least one edge 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 graph
disjoint union of the cycle graph 
and the claw graph 
is isomorphic to the graph
disjoint union 
of two triangle graphs,
which is vertex-transitive, while
the original graph is not edge-transitive.
Counting empty graphs as edge-transitive, the numbers of edge-transitive unlabeled graphs on 
, 2, ... nodes are 1, 2, 4, 8, 12, 21, 26, 38, 49, 67, ...
(OEIS A095352).
The numbers of connected edge-transitive unlabeled graphs on 
,
2, ... nodes are 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, ... (OEIS A095424;
cf. Conder 2017).
MathWorld calls a graph that is both edge-transitive and vertex-transitive a symmetric graph (Holton and Sheehan 1993, pp. 209-210); see that entry for alternative terminology. A regular graph that is edge-transitive but not vertex-transitive is called a semisymmetric graph.
