What does a dimension of a homology group tell us? In particular, suppose we form an arbitrary simplicial complex $S(G)$ from a simple graph $G$. Then we compute the homology groups of $S(G)$ and note the ones which are nonzero. What useful information can we glean by computing the dimension of the nonzero homology groups?
Edit. $S(G)$ is usually taken to be the coloring complex.