Say we have a graph $G=(V(G),E(G))$, where $V(G)$ is the vertex set of $G$ and $E(G)$ is the edge set of $G$. Define a graph $H$ which preserves all adjacency relations, $V(G)=E(H)$ and $V(H)=E(G)$. Is there a name for $H$?
I should mention that I am not entirely sure that the previous definition is correct, so here's an intuitive explanation on what I'm thinking of. For example, say we have $G=C_k$ (the cycle of order $k$), so that $V(G)=\{v_1,v_2,\ldots,v_k\}$ and $E$ consists of $v_iv_{i+1}$ (with indices taken modulo $k$). Then the graph $H$ in this case would be again just $C_k$ with different vertex/edge labellings.
As another example, the path of order two $G=P_2$ would have a corresponding graph $H$ which has one vertex (since $G$ has one edge) and two edges (since $G$ has two vertices). In this case, however, I'm not sure that $H$ is well defined since it's not clear what two vertices these two edges are incident with.
So, is this a well-established concept for most graphs? When is it not well defined? Is there a name for this graph $H$ corresponding to $G$? Thanks in advance.
Your definition of $E(H)$ runs into problems with non-degree-2 vertices in $G$, as mentioned in comments. But if this change is made to the definition of $E(H)$:
then $H$ is the line graph of $G$.