Torsion criterion for graph cohomology of algebras ${\mathcal A}_m$

Let $G$ be a graph, and let $H^{**}_{{\mathcal A}_m}(G)$ denote its graph cohomology associated with the algebra ${\mathcal A}_m$. A graph is loopless if it has no loops, and a cycle has order equal to its number of edges. Torsion criterion for ${\mathcal A}_m$. The cohomology $H^{**}_{{\mathcal A}_m}(G)$ contains torsion if and only if $G$ is loopless and contains a cycle of order at least $3$. In this case, $H^{**}_{{\mathcal A}_m}(G)$ has torsion of order dividing $m$. This generalizes the corresponding result for ${\mathcal A}_2$; the source later notes that this conjecture was proved in the cited work.