Minimal Path and Acyclic Models in the Path Complex

In this paper, firstly, we will study the structure of the path complex $({\mathrm{\Omega }}_{\ast }(G;\mathbb{Z}),\mathrm{\partial })$ of a digraph $G$ via the $\mathbb{Z}$-generators of ${\mathrm{\Omega }}_{\ast }(G,\mathbb{Z})$ under strongly regular condition, which is called the minimal path in \cite{HY}. In particular, we will study various examples of the minimal $3$-paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model.