Differential algebras on digraphs and parametrized homology

TL;DR

This paper introduces a new framework called parametrized homology for directed graphs, extending existing path homology, and proves its functoriality and related Kunneth-type formulas.

Contribution

It defines parametrized homology for digraphs as an analog of path homology and establishes its fundamental properties including functoriality and Kunneth-type formulas.

Findings

Parametrized homology is functorial for digraphs.

Kunneth-type formulas are established for the homology.

The framework generalizes path homology to a parametrized setting.

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.