Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces. Formally, the nth Betti number is the rank of the nth homology group of a topological space. The first Betti number of a graph is commonly known as its circuit rank (or cyclomatic number). The following table gives the Betti number of some common surfaces.

chromatic number | circuit rank | Euler characteristic | genus | homology group | topological space