Editing Cyclomatic number (section)

==The number of independent cycles==
In [[algebraic graph theory]], the cyclomatic number is also the dimension of the [[cycle space]] of <math>G</math>. Intuitively, this can be explained as meaning that the cyclomatic number counts the number of independent cycles in the graph, where a collection of cycles is independent if it is not possible to form one of the cycles as the [[symmetric difference]] of some subset of the others.<ref name="berge"/>

This count of independent cycles can also be explained using [[homology theory]], a branch of topology. Any graph {{mvar|G}} may be viewed as an example of a {{nowrap|1-dimensional}} [[simplicial complex]], a type of topological space formed by representing each graph edge by a [[line segment]] and gluing these line segments together at their endpoints.
The cyclomatic number is the [[Rank of an abelian group|rank]] of the first ([[integer]]) [[homology group]] of this complex,<ref>{{citation|title=Trees|first=Jean-Pierre|last=Serre|authorlink=Jean-Pierre Serre|page=23|publisher=Springer|series=Springer Monographs in Mathematics|year=2003|isbn=9783540442370 |url=https://books.google.com/books?id=MOAqeoYlBMQC&pg=PA23}}.</ref>
<math display=block>r = \operatorname{rank}\left[H_1(G,\Z)\right].</math>
Because of this topological connection, the cyclomatic number of a graph {{mvar|G}} is also called the '''first [[Betti number]]''' of {{mvar|G}}.<ref name="BerkolaikoKuchment2013">{{citation|author1=Gregory Berkolaiko|author2=Peter Kuchment|title=Introduction to Quantum Graphs|url=https://books.google.com/books?id=QAs8tiBsvEoC&pg=PA4|year=2013|publisher=American Mathematical Soc.|isbn=978-0-8218-9211-4|pages=4}}</ref> More generally, the first Betti number of any topological space, defined in the same way, counts the number of independent cycles in the space.