# Is there a name for a graph whose vertices consist of edges of an existing graph connected by an edge if they shared a common vertex?

I was wondering if there is a name for this construction: Take a graph $$G$$ and construct a new graph $$G'$$ in which the edges of $$G$$ now become vertices, and "vertices become edges" in the sense that vertices are joined by edge in $$G'$$ if they as edges in $$G$$ shared a common vertex.

More precisely: Given a graph $$G=(V=\{v_1,v_2,\dots,v_n\},E=\{e_1,e_2,\dots,e_k\})$$ where each $$e_i=\{u,v\}\subset V$$ we define the graph $$G'=(V'=\{e_1,e_2,\dots,e_k\},E')$$ so that $$\{e_i,e_j\}\in E' \iff e_i\cap e_j\neq\emptyset$$.

I don't know if this can be used for anything (my guess would be it can't) it just came to my mind and thought that perhaps this has a name.

This is called the line graph of $$G$$. It actually has a wide variety of uses, as seen on that Wikipedia page, and the terminology is sufficiently standard that you should be able to mention it in a paper without defining it.