Discrete Cubical and Path Homologies of Graphs

TL;DR

This paper compares two homology theories for graphs, showing they are isomorphic in some cases but differ in higher dimensions, and develops tools for computing these homologies.

Contribution

It provides a detailed comparison of cubical and path homologies for graphs, including conditions for isomorphism and tools for computation.

Findings

Homology groups are isomorphic in dimension one for both theories.

Counterexamples show differences in higher dimensions.

A natural map between the theories is an isomorphism in dimension one and surjective in dimension two.

Abstract

In this paper we study and compare two homology theories for (simple and undirected) graphs. The first, which was developed by Barcelo, Caprano, and White, is based on graph maps from hypercubes to the graph. The second theory was developed by Grigor'yan, Lin, Muranov, and Yau, and is based on paths in the graph. Results in both settings imply that the respective homology groups are isomorphic in homological dimension one. We show that, for several infinite classes of graphs, the two theories lead to isomorphic homology groups in all dimensions. However, we provide an example for which the homology groups of the two theories are not isomorphic at least in dimensions two and three. We establish a natural map from the cubical to the path homology groups which is an isomorphism in dimension one and surjective in dimension two. Again our example shows that in general the map is not surjective in dimension three and not injective in dimension two. In the process we develop tools to compute the homology groups for both theories in all dimensions.