The Cellular Homology of Digraphs

In \cite{TY}, we investigate the pair $(P,\text{Supp}(P))$ of minimal path $P$ and its supporting sub-digraph $\text{Supp}(P)$ in the path complex of a digraph $G$ under the strongly regular condition. In this paper, first, we consider the special minimal path $P$ specified by the admissible condition (Definition \ref{admpair}), which means that $(P,\text{Supp}(P))$ admits a singular cubical realization. Based on such a subset, we systematically introduce the definitions of cellular chain complex associated to $G$ and prove the well-definedness. Then we study several properties of such cellular homologies. Finally, we present several intriguing examples as well as some important observations.