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 >= 2. We also construct a sequence {G(d)} of graphs such that this homology is nontrivial in dimension d for d >= 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.