For any simple, undirected graph $G$, let $L(G)$ denote its 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 $G\cong L(G)$?
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.
