  In this paper we study and compare two homology theories for (simple and
undirected) graphs. The first, which was developed by Barcelo, Caprano, and
White, is based on graph maps from hypercubes to the graph. The second theory
was developed by Grigor'yan, Lin, Muranov, and Yau, and is based on paths in
the graph. Results in both settings imply that the respective homology groups
are isomorphic in homological dimension one. We show that, for several infinite
classes of graphs, the two theories lead to isomorphic homology groups in all
dimensions. However, we provide an example for which the homology groups of the
two theories are not isomorphic at least in dimensions two and three. We
establish a natural map from the cubical to the path homology groups which is
an isomorphism in dimension one and surjective in dimension two. Again our
example shows that in general the map is not surjective in dimension three and
not injective in dimension two. In the process we develop tools to compute the
homology groups for both theories in all dimensions.
