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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.

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    @PEV: that's only one of many equivalent definitions of the Euler characteristic. There's one that relates it directly to cell counts in your simplicial complex. If you go on to apply the definition of your simplicial complex you get a formula for your Euler characteristic directly in terms of the properties of $G$.2011-05-28

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