I am interested in the intersection graphs of $\binom{X}{2}$, i.e. the set of all 2-element subsets of a (finite) set $X$.


[**Motivation**: One can represent every simple graph with $n$ vertices by an assignment of 0 or 1 to the vertices of the intersection graph of $\binom{[n]}{2}$. This is a somehow more "natural" (= "coordinate free") representation of a graph than the usual adjacency matrix.]


Especially I wonder



> 
> * In which (other) contexts are/were these graphs investigated?
> * How can they be characterized abstractly?
> * And do they have a name, thus being characterized?
> 
> 
> 



Intersection graphs of 2-element subsets
The usual name of your graph is $L(K\_n)$, the **line graph** of the complete graph $K\_n$. The line graphs of complete graphs have also been called **triangular** graphs, as I just learned moments ago from the Wikipedia article <http://en.wikipedia.org/wiki/Line_graph> .


