Differential algebras on digraphs and parametrized homology

Abstract: 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 consider the differential algebras on digraphs and define the parametrized homology of digraphs as an analog of the path homology. We prove the functoriality of the parametrized homology of digraphs in Theorem 4.2. We also prove some Kunneth-type formulae for the parametrized homology of digraphs in Theorem 4.4.