What is a graph where edges are also vertices ? Is there a name for a kind of graph where edges are vertices in the same graph ? 
A example would be :

e1(a,b)
  e2(c,d)
  e3(e1,e)

 A: Not sure but in RDF you have that.  They are called labelled directed graphs in the "RDF Concepts" spec and directed graphs in RDF Semantics.
For example in Turtle - which is just one notation for RDF Graphs - you can write
@prefix foaf: <http://xmlns.com/foaf/0.1/>.
@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>.

rdf:type a rdf:Property .
<http://bblfish.net/#hjs> rdf:type foaf:Person .

Here the rdf:type name is in subject position in the first statement (a vertice?) and in predicate position (an edge) in the second sentence .
A: Updated Answer
I am not sure if this is exactly what you are looking for, but one way to have the edges as nodes in the same graph is using Bipartite graphs:
http://en.wikipedia.org/wiki/Bipartite_graph
For example,
$$G = (V,E) = (\{ v_1, v_2 \}, \{ (v_1,v_2) \}).$$
Can be represented by,
$$G' = (V',U,E') = (\{ v_1, v_2 \}, \{ (v_1,v_2) \}, \{ (v_1,(v_1,v_2)), (v_2,(v_1,v_2))\}).$$
Where $(v_1,v_2)$ is now a node, and nodes $v_1$,$v_2$ is connected to it.
Previous Wrong Answer
They are sometimes called "dual graphs" (e.g. http://people.hofstra.edu/geotrans/eng/methods/dual_graph.html). And sometimes called "edge dual graphs".
However, I understand that "dual graphs" can also refer to the dual graphs of planar graphs.
