Differential Algebras on Digraphs and Generalized Path Homology.

The theory of path homology for digraphs was developed by Alexander Grigor'yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. In this paper, we generalize the path homology for digraphs. We prove that for any digraph $G$, any $t\geq 0$, any $0\leq q\leq 2t$, and any $(2t+1)$-dimensional element $\alpha$ in the differential algebra on the set of the vertices, we always have an $(\alpha,q)$-path homology for $G$. In particular, if $t=0$, then the $(\alpha,0)$-path homology gives the weighted path homology for vertex-weighted digraphs.