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 $G\cong L(G)$?