  In this paper, firstly, we will study the structure of the path complex
$(\Omega_*(G)_{\mathbb{Z}},\partial)$ via the $\mathbb{Z}$-generators of
$\Omega_*(G)_{\mathbb{Z}}$, which is called the minimal path in \cite{HY}. In
particular, we will study various examples of the minimal paths of length 3.
Secondly, we will show that the supporting graph of minimal graph is acyclic in
the path homology. Thirdly, we will consider the applications of the acyclic
models.
