# Is there a name for this type of graphs?

From a graph G I want to construct a graph (lets call it) G# with the following properties:

• Each node and each edge of G is a node of G#
• For each e in G connecting the nodes n1, n2 there exist the edges connecting e,n1 and e,n2
• no other edges exist in G#

G# is then bipartite with the nodes and edges of G being it's disjoint sets´. It's like creating the line graph of G and stopping after the creation of the "new nodes".

Is there a name for this special kind of graph?

This is more like the Total Graph than the Line Graph (because the nodes of $G\#$ consist of both the nodes and the edges of $G$), except that adjacency in $G\#$ is determined only by incidence in $G$, rather than both incidence and adjacency in $G$ (which is the case in Line Graphs).