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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.

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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.

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