Let $G$ be locally finite graph. By $|G|$, we mean the Freudenthal compactification of the 1-complex $G$, see chapter 8 of "Graph Theory" by Diestel. It is followed from Lemma 8.5.4 of Diestel's book: Every connected standard subspace of $|G|$ is arc-connected. So if $|G|$ is a connected space, then $|G|$ is an arc-connected space, and so $G$ is connected as graph.
Now, my question is: Assume that $G$ is connected as graph. Is $|G|$ is a connected space?
It is true in the finite case. But I am not sure about the locally finite case.