For any simple, undirected graph $G$, let $L(G)$ denote its [line graph](https://en.wikipedia.org/wiki/Line_graph). 


$G=(\mathbb{Z}, E)$ with $E = \{\{k, k+1\}:k\in \mathbb{Z}\}$ has the property that $G\cong L(G)$.


Is there a connected infinite graph $G$ such that every vertex has more than $2$ neighbors, and $G\cong L(G)$?



Infinite connected graphs isomorphic to their line graph
There is an embedding of the complete graph on countably many vertices $K\_{\aleph\_0}$ into its line graph $LK\_{\aleph\_0}$ as the clique associated to one of the vertices. This induces embeddings $L^n K\_{\aleph\_0} \to L^{n+1}K\_{\aleph\_0}$ for all $n$. The iterated union of $L^n K\_{\aleph\_0}$ under all these embeddings has the desired property.


