The homology of digraphs as a generalisation of Hochschild homology

TL;DR

This paper introduces a new homology theory for directed graphs with coefficients in arbitrary bimodules, extending Hochschild homology to non-commutative algebras and generalizing previous graph homology results.

Contribution

It defines a novel homology for directed graphs with non-commutative coefficients, generalizing Hochschild homology and connecting it to graph homology.

Findings

The new homology agrees with Hochschild homology on polygons in certain dimensions.

It extends graph homology to non-commutative coefficient rings.

Provides a framework for analyzing directed graphs with complex algebraic structures.

Abstract

J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a new homology theory for directed graphs which takes coefficients in an arbitrary $A-A$ bimodule, for $A$ possibly non-commutative, which on polygons agrees with Hochschild homology through a range of dimensions.