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 connectedundirected 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).