Editing Cycle space (section)

===Algebra===
If one applies any [[Set (mathematics)#Basic operations|set operation]] such as union or intersection of sets to two spanning subgraphs of a given graph, the result will again be a subgraph. In this way, the edge space of an arbitrary graph can be interpreted as a [[Boolean algebra (structure)|Boolean algebra]].<ref>{{citation|title=Applied Discrete Structures|first=K. D.|last=Joshi|publisher=New Age International|year=1997|isbn=9788122408263|page=172|url=https://books.google.com/books?id=lxIgGGJXacoC&pg=PA172}}.</ref>

[[File:Cycle space addition.svg|thumb|360px|The symmetric difference of two Eulerian subgraphs (red and green) is a Eulerian subgraph (blue).]]
The cycle space, also, has an algebraic structure, but a more restrictive one. The union or intersection of two Eulerian subgraphs may fail to be Eulerian. However, the [[symmetric difference]] of two Eulerian subgraphs
(the graph consisting of the edges that belong to exactly one of the two given graphs) is again Eulerian.<ref name="gy"/> This follows from the fact that the symmetric difference of two sets with an even number of elements is also even. Applying this fact separately to the [[neighbourhood (graph theory)|neighbourhood]]s of each vertex shows that the symmetric difference operator preserves the property of being Eulerian.

A family of sets closed under the symmetric difference operation can be described algebraically as a [[vector space]] over the two-element [[finite field]] [[GF(2)|<math>\Z_2</math>]].<ref>{{citation|title=A Beginner's Guide to Graph Theory|first=W. D.|last=Wallis|publisher=Springer|year=2010|isbn=9780817645809|page=66|url=https://books.google.com/books?id=240QO32GJOcC&pg=PA66}}.</ref> This field has two elements, 0 and 1, and its addition and multiplication operations can be described as the familiar addition and multiplication of [[integer]]s, taken [[Modular arithmetic|modulo&nbsp;2]]. A vector space consists of a set of elements together with an addition and scalar multiplication operation satisfying certain properties generalizing the properties of the familiar [[real vector space]]s. For the cycle space, the elements of the vector space are the Eulerian subgraphs, the addition operation is symmetric differencing, multiplication by the scalar&nbsp;1 is the [[identity operation]], and multiplication by the scalar&nbsp;0 takes every element to the empty graph, which forms the [[additive identity]] element for the cycle space.

The edge space is also a vector space over <math>\Z_2</math> with the symmetric difference as addition. As vector spaces, the cycle space and the [[cut space]] of the graph (the family of edge sets that span the [[Cut (graph theory)|cuts]] of the graph) are the [[orthogonal complement]]s of each other within the edge space. This means that a set <math>S</math> of edges in a graph forms a cut if and only if every Eulerian subgraph has an even number of edges in common with <math>S</math>, and <math>S</math> forms an Eulerian subgraph if and only if every cut has an even number of edges in common with <math>S</math>.<ref name="diestel"/> Although these two spaces are orthogonal complements, some graphs have nonempty subgraphs that belong to both of them. Such a subgraph (an Eulerian cut) exists as part of a graph <math>G</math> if and only if <math>G</math> has an even number of [[spanning forest]]s.<ref>{{citation|first=David|last=Eppstein|authorlink=David Eppstein|title=On the Parity of Graph Spanning Tree Numbers|year=1996|url=http://www.ics.uci.edu/~eppstein/pubs/Epp-TR-96-14.pdf|series=Technical Report 96-14|publisher=Department of Information and Computer Science, University of California, Irvine}}.</ref>