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On the vanishing of discrete singular cubical homology for graphs

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.