Editing Cycle space (section)

===Topology===
An undirected graph may be viewed as a [[simplicial complex]] with its vertices as zero-dimensional simplices and the edges as one-dimensional simplices.<ref name="serre">{{citation|title=Trees|first=Jean-Pierre|last=Serre|authorlink=Jean-Pierre Serre|page=23|publisher=Springer|series=Springer Monographs in Mathematics|year=2003|url=https://books.google.com/books?id=MOAqeoYlBMQC&pg=PA23|isbn=9783540442370}}.</ref> The [[chain complex]] of this topological space consists of its edge space and [[vertex space]] (the Boolean algebra of sets of vertices), connected by a boundary operator that maps any spanning subgraph (an element of the edge space) to its set of odd-degree vertices (an element of the vertex space). The [[homology group]]
:<math>H_1(G,\Z_2)</math>
consists of the elements of the edge space that map to the zero element of the vertex space; these are exactly the Eulerian subgraphs. Its group operation is the symmetric difference operation on Eulerian subgraphs.

Replacing <math>\Z_2</math> in this construction by an arbitrary [[ring (mathematics)|ring]] allows the definition of cycle spaces to be extended to cycle spaces with coefficients in the given ring, that form [[module (mathematics)|module]]s over the ring.<ref>{{citation|title=Algebraic Graph Theory|series=Cambridge Mathematical Library|first=Norman|last=Biggs|publisher=Cambridge University Press|year=1993|isbn=9780521458979|page=154|url=https://books.google.com/books?id=6TasRmIFOxQC&pg=PA154}}.</ref>
In particular, the '''integral cycle space''' is the space
:<math>H_1(G,\Z).</math>
It can be defined in graph-theoretic terms by choosing an arbitrary [[orientation (graph theory)|orientation]] of the graph, and defining an '''integral cycle''' of a graph <math>G</math> to be an assignment of integers to the edges of <math>G</math> (an element of the [[free abelian group]] over the edges) with the property that, at each vertex, the sum of the numbers assigned to incoming edges equals the sum of the numbers assigned to outgoing edges.<ref name="mcba">{{citation|title=Algorithmics of Large and Complex Networks|series=Lecture Notes in Computer Science|volume=5515|year=2009|pages=34–49|contribution=Minimum cycle bases and their applications|first1=Franziska|last1=Berger|first2=Peter|last2=Gritzmann|first3=Sven|last3=de Vries|doi=10.1007/978-3-642-02094-0_2|isbn=978-3-642-02093-3}}.</ref>

A member of <math>H_1(G,\Z)</math> or of <math>H_1(G,\Z_k)</math> (the cycle space modulo <math>k</math>) with the additional property that all of the numbers assigned to the edges are nonzero is called a [[nowhere-zero flow]] or a nowhere-zero <math>k</math>-flow respectively.<ref>{{citation
 | last = Seymour | first = P. D. | authorlink = Paul Seymour (mathematician)
 | contribution = Nowhere-zero flows
 | location = Amsterdam
 | mr = 1373660
 | pages = 289–299
 | publisher = Elsevier
 | title = Handbook of combinatorics, Vol. 1, 2
 | year = 1995}}.</ref>