Let $\Delta$ be an abstract simplicial complex https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex and $|\Delta|$ be its geometric realization.
Let $\Delta ^{(1)} :=\{F $ is a face of $\Delta : |F|\le 2 \}$ . So $\Delta ^{(1)}$ is a simplicial complex of dimension $1$ , so in particular it is a simple graph. Now $\Delta ^{(1)}$ is connected as a graph iff its geometric realization $|\Delta ^{(1)}|$ is path connected. Now consider the following two statements:
(1) $\Delta ^{(1)}$ is connected as a graph.
(2) $|\Delta |$ is path connected as a topological space.
My question is: are (1) and (2) equivalent ? If not then at least, does one of them imply the other ?