On the vanishing of discrete singular cubical homology for graphs

TL;DR

This paper investigates conditions under which the discrete cubical homology of graphs vanishes or remains non-trivial, revealing connections to graph cycles and associated cell complexes.

Contribution

It characterizes when the discrete cubical homology of graphs is trivial or non-trivial and relates it to classical homology of constructed cell complexes.

Findings

Graphs without 3- and 4-cycles have trivial homology in dimensions ≥ 2.

Constructed graphs with certain properties have non-trivial homology in all dimensions ≥ 1.

Discrete cubical homology can differ from singular homology depending on coverings and associated complexes.

Abstract

We prove that if G is a graph without 3-cycles and 4-cycles, then the discrete cubical homology of G is trivial in dimension d, for all d\ge 2. We also construct a sequence { G_d } of graphs such that this homology is non-trivial in dimension d for d\ge 1. Finally, we show that the discrete cubical homology induced by certain coverings of G equals the ordinary singular homology of a 2-dimensional cell complex built from G, although in general it differs from the discrete cubical homology of the graph as a whole.