Editing Cycle space (section)

==Circuit rank==
{{main|Circuit rank}}
As a vector space, the dimension of the cycle space of a graph with <math>n</math> vertices, <math>m</math> edges, and <math>c</math> [[Connected component (graph theory)|connected components]] is <math>m-n+c</math>.<ref name="gy"/><ref name="diestel"/><ref name="berge">{{citation|title=The Theory of Graphs|first=Claude|last=Berge|authorlink=Claude Berge|publisher=Courier Dover Publications|year=2001|isbn=9780486419756|pages=27–30|contribution=Cyclomatic number|url=https://books.google.com/books?id=h5BjnaoKyOwC&pg=PA27}}.</ref> This number can be interpreted topologically as the first [[Betti number]] of the graph.<ref name="serre"/> In graph theory, it is known as the [[circuit rank]], cyclomatic number, or nullity of the graph.

Combining this formula for the rank with the fact that the cycle space is a vector space over the two-element field shows that the total number of elements in the cycle space is exactly <math>2^{m-n+c}</math>.