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In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings.

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rdfs:label	Cycle space (en) Пространство циклов (ru) Простір циклів (uk)
rdfs:comment	In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings. (en) Про́стір ци́клів (або цикловий простір) неорієнтованого графа — лінійний простір над полем , що складається з його ейлерових підграфів. Розмірність цього простору називають контурним рангом графа. З точки зору алгебричної топології цикловий простір є першою групою гомологій графа. (uk) Пространство циклов неориентированного графа — линейное пространство над полем , состоящее из его эйлеровых подграфов. Размерность этого пространства называется контурным рангом графа. С точки зрения алгебраической топологии циклическое пространство является первой группой гомологий графа. (ru)
foaf:depiction
dct:subject	Algebraic graph theory
Wikipage page ID	262243 (xsd:integer)
Wikipage revision ID	1117466378 (xsd:integer)
Link from a Wikipage to another Wikipage	Saunders Mac Lane Module (mathematics) Nowhere-zero flow Mac Lane's planarity criterion Basis (linear algebra) Algebraic topology Betti number Cycle (graph theory) Cycle basis Undirected graph Vector space Degree (graph theory) Induced path Complement (set theory) Connected graph Neighbourhood (graph theory) Orientation (graph theory) GF(2) Glossary of graph theory Gomory–Hu tree Graph coloring Modular arithmetic NP-hard Leonhard Euler Additive identity Algebraic graph theory K-vertex-connected graph Cut (graph theory) Cut space Dual graph Edge space Even number Finite field Four color theorem Graph theory Simplicial complex Ring (mathematics) Snark theorem Homology (mathematics) Identity operation Boolean algebra (structure) Planar graph Circuit rank Free abelian group Connected component (graph theory) Real vector space Empty graph Integers Orthogonal complement Chain complex Set (mathematics) Vertex (graph theory) Symmetric difference Seven Bridges of Königsberg Spanning forest Veblen's theorem Eulerian graph Vertex space Homology group
sameAs	Cycle space Cycle space Cycle space Cycle space Cycle space Cycle space
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thumbnail	wiki-commons:Special:FilePath/Cycle_space_addition.svg?width=300
has abstract	In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings. (en) Про́стір ци́клів (або цикловий простір) неорієнтованого графа — лінійний простір над полем , що складається з його ейлерових підграфів. Розмірність цього простору називають контурним рангом графа. З точки зору алгебричної топології цикловий простір є першою групою гомологій графа. (uk) Пространство циклов неориентированного графа — линейное пространство над полем , состоящее из его эйлеровых подграфов. Размерность этого пространства называется контурным рангом графа. С точки зрения алгебраической топологии циклическое пространство является первой группой гомологий графа. (ru)
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prov:wasDerivedFrom	wikipedia-en:Cycle_space?oldid=1117466378&ns=0
page length (characters) of wiki page	17848 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Cycle_space
is Link from a Wikipage to another Wikipage of	Nowhere-zero flow Mac Lane's planarity criterion List of graph theory topics Cycle (graph theory) Cycle basis Incidence matrix Petersen graph Glossary of graph theory Spanning tree Peripheral cycle W. T. Tutte Cut (graph theory)

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