SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 35-54, January 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.

中文翻译：

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关于图离散奇异三次同调的消失

SIAM Journal on Discrete Mathematics，第 35 卷，第 1 期，第 35-54 页，2021 年 1 月。
我们证明如果 $G$ 是一个没有 3-cycles 和 4-cycles 的图，那么 $G$ 的离散三次同调是微不足道的在所有 $d\ge 2$ 的维度 $d$ 中。我们还构建了一个序列 $\{G_d\}$ 的图，使得这种同源性在维度 $d$ 上对于 $d\ge 1$ 是非平凡的。最后，我们表明由 $G$ 的某些覆盖引起的离散立方同调等于由 $G$ 构建的 $2$ 维细胞复合体的普通奇异同调，尽管通常它不同于图的离散立方同调为整个。