I have a loop-free graph $G$ with vertex set $V$ and edges $E$. I now construct a graph $G'$ which has as "vertex" set $E$ and edges between two "vertices" if the corresponding edges in $G$ have a common end vertex $v\in V$.
Is there a name for $G'$ or is there a more "canonical way" to assign edges in $G'$ to retain as much structure from $G$ as possible?
I am looking for something like a dual graph but with the edges and not the faces.
EDIT: It's called line graph or adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange or representative graph ( see http://mathworld.wolfram.com/LineGraph.html)
It's called the "dual graph" of $G$ IIRC.
Edit I remembered wrongly: Diestel's book calls it the line graph $L(G)$ of $G$. And Bóllobás' book as well. Dual is used for planar graphs instead.