The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).

There are countable graphs that are isomorphic to their line graph:

• $G=(\omega,E)$ where $E=\{\{k,k+1\}: k\in\omega\}$;
• $G=(\mathbb{Z},E)$ where $E=\{\{k,k+1\}: k\in\mathbb{Z}\}$.

Note that in the second graph, all vertices have degree 2, which makes it into a kind of "infinite cycle". (An interesting side question would be whether these are the only countable graphs (up to isomorphism) that are isomorphic to their line graph.)

Question. Is there a connected graph $G=(V,E)$ with $V$ uncountable such that $G\cong L(G)$?