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== Cycle space 环空间 ==

== Cycle space 环空间 ==

The term ''cycle'' may also refer to an element of the [[cycle space]] of a graph. There are many cycle spaces, one for each coefficient field or ring.  The most common is the ''binary cycle space'' (usually called simply the ''cycle space''), which consists of the edge sets that have even degree at every vertex; it forms a [[vector space]] over the two-element [[finite field|field]].  By [[Veblen's theorem]], every element of the cycle space may be formed as an edge-disjoint union of simple cycles. A [[cycle basis]] of the graph is a set of simple cycles that forms a [[basis (linear algebra)|basis]] of the cycle space.<ref name="gy">{{citation|title=Graph Theory and Its Applications|edition=2nd|first1=Jonathan L.|last1=Gross|first2=Jay|last2=Yellen|publisher=CRC Press|year=2005|isbn=9781584885054|chapter=4.6 Graphs and Vector Spaces|pages=197–207|url=https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA197}}.</ref>

The term ''cycle'' may also refer to an element of the [[cycle space]] of a graph. There are many cycle spaces, one for each coefficient field or ring.  The most common is the ''binary cycle space'' (usually called simply the ''cycle space''), which consists of the edge sets that have even degree at every vertex; it forms a [[vector space]] over the two-element [[finite field|field]].  By [[Veblen's theorem]], every element of the cycle space may be formed as an edge-disjoint union of simple cycles. A [[cycle basis]] of the graph is a set of simple cycles that forms a [[basis (linear algebra)|basis]] of the cycle space.

The term cycle may also refer to an element of the cycle space of a graph. There are many cycle spaces, one for each coefficient field or ring.  The most common is the binary cycle space (usually called simply the cycle space), which consists of the edge sets that have even degree at every vertex; it forms a vector space over the two-element field.  By Veblen's theorem, every element of the cycle space may be formed as an edge-disjoint union of simple cycles. A cycle basis of the graph is a set of simple cycles that forms a basis of the cycle space.

The term cycle may also refer to an element of the cycle space of a graph. There are many cycle spaces, one for each coefficient field or ring.  The most common is the binary cycle space (usually called simply the cycle space), which consists of the edge sets that have even degree at every vertex; it forms a vector space over the two-element field.  By Veblen's theorem, every element of the cycle space may be formed as an edge-disjoint union of simple cycles. A cycle basis of the graph is a set of simple cycles that forms a basis of the cycle space.

关于术语“'''<font color="#ff8000"> 环Cycle</font>'''”还可以指代一个图中环空间的一个元素。在一个图中，存在很多环空间，每个都有对应的'''<font color="#ff8000"> 系数域Coefficient field</font>'''或'''<font color="#ff8000"> 环Ring（代数）</font>'''。最常见的是二元环空间（通常简称为环空间），它是由在该图中每个顶点上具有偶数度的边集组成。它在二元域上形成了一个向量空间。根据'''<font color="#ff8000"> 维布伦定理Veblen's theorem</font>'''，该环空间的每个元素都可以形成为简单环的不相交边的并集。该图的'''<font color="#ff8000"> 环基Cycle basis</font>'''相当于一组简单环，它们构成了环空间的基。

关于术语“'''<font color="#ff8000"> 环Cycle</font>'''”还可以指代一个图中'''<font color="#ff8000"> 环空间Cycle space</font>'''的一个元素。在一个图中，存在很多环空间，每个都有对应的'''<font color="#ff8000"> 系数域Coefficient field</font>'''或'''<font color="#ff8000"> 环Ring（代数）</font>'''。最常见的是'''<font color="#ff8000"> 二元环空间Binary cycle space</font>'''（通常简称为环空间），它是由在该图中每个顶点上具有偶数度的边集组成。它在二元域上形成了一个'''<font color="#ff8000"> 向量空间Vector space</font>'''。根据'''<font color="#ff8000"> 维布伦定理Veblen's theorem</font>'''，该环空间的每个元素都可以形成为简单环的不相交边的并集。该图的'''<font color="#ff8000"> 环基Cycle basis</font>'''相当于一组简单环，它们构成了环空间的基。