On the Vanishing of Discrete Singular Cubical Homology for Graphs
Hélène Barcelo, Curtis Greene, Abdul Salam Jarrah, Volkmar Welker · SIAM Journal on Discrete Mathematics · 2021
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 nontrivial 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.