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Moved page from wikipedia:en:Incidence matrix (history)
此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。
In [[mathematics]], an '''incidence matrix''' is a [[Matrix (mathematics)|matrix]] that shows the relationship between two classes of objects. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element of ''X'' and one column for each element of ''Y''. The entry in row ''x'' and column ''y'' is 1 if ''x'' and ''y'' are related (called ''incident'' in this context) and 0 if they are not. There are variations; see below.
In mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 if x and y are related (called incident in this context) and 0 if they are not. There are variations; see below.
在数学中，关联矩阵是表示两类对象之间关系的矩阵。如果第一个类是 x，第二个类是 y，那么这个矩阵对于 x 的每个元素有一行，对于 y 的每个元素有一列。如果 x 和 y 是相关的(在本文中称为 incident) ，则行 x 和列 y 中的条目为1，如果它们不是相关的，则为0。这里有一些变化，请看下面。
==Graph theory==
Incidence matrices are frequently used in [[graph theory]].
Incidence matrices are frequently used in graph theory.
关联矩阵是图论中经常使用的一种方法。
===Undirected and directed graphs===
[[Image:Labeled undirected graph.svg|thumb|250px|An undirected graph.]]
An undirected graph.
无向图一个无向图。
In graph theory an [[undirected graph]] has two kinds of incidence matrices: unoriented and oriented.
In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented.
在图论中，无向图有两种关联矩阵: 无向图和有向图。
The ''unoriented incidence matrix'' (or simply ''incidence matrix'') of an undirected graph is a {{nowrap|''n'' × ''m''}} [[Matrix (math)|matrix]] ''B'', where ''n'' and ''m'' are the numbers of [[Vertex (graph theory)|vertices]] and [[Edge (graph theory)|edge]]s respectively, such that {{nowrap|1=''B''<sub>''i'',''j''</sub> = 1}} if the vertex ''v''<sub>''i''</sub> and edge ''e''<sub>''j''</sub> are incident and 0 otherwise.
The unoriented incidence matrix (or simply incidence matrix) of an undirected graph is a matrix B, where n and m are the numbers of vertices and edges respectively, such that if the vertex v<sub>i</sub> and edge e<sub>j</sub> are incident and 0 otherwise.
无向图的无向关联矩阵是一个矩阵 b，其中 n 和 m 分别是顶点和边的数目，如果顶点 v < sub > i </sub > 和边 e < sub > j </sub > 是关联的，否则为0。
For example the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1–4) and 4 columns (corresponding to the four edges, e1–e4):
For example the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1–4) and 4 columns (corresponding to the four edges, e1–e4):
例如，右边所示的无向图的关联矩阵是一个由4行(对应4个顶点，1-4)和4列(对应4条边，e1-e4)组成的矩阵:
{|
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{| align=left class=wikitable
{| align=left class=wikitable
{ | align = left class = wikable
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! !! e<sub>1</sub> !! e<sub>2</sub> !! e<sub>3</sub> !! e<sub>4</sub>
! !! e<sub>1</sub> !! e<sub>2</sub> !! e<sub>3</sub> !! e<sub>4</sub>
!!!电子邮件！2! ！3 </sub > ！E < sub > 4 </sub >
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!1
!1
!1
|1||1||1||0
|1||1||1||0
|1||1||1||0
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!2
!2
|1||0||0||0
|1||0||0||0
|1||0||0||0
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!3
!3
|0||1||0||1
|0||1||0||1
|0||1||0||1
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|0||0||1||1
|0||0||1||1
|0||0||1||1
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<math>
<math>
《数学》
\begin{pmatrix}
\begin{pmatrix}
开始{ pmatrix }
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
\end{pmatrix}.
\end{pmatrix}.
结束{ pmatrix }。
</math>
</math>
数学
|}
|}
|}
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.
如果我们看一下关联矩阵，我们会发现每一列的总和等于2。这是因为每条边都有一个顶点连接到每个端点。
The ''incidence matrix'' of a [[directed graph]] is a {{nowrap|''n'' × ''m''}} matrix ''B'' where ''n'' and ''m'' are the number of vertices and edges respectively, such that {{nowrap|1=''B''<sub>''i'',''j''</sub> = −1}} if the edge ''e''<sub>''j''</sub> leaves vertex ''v''<sub>i</sub>, 1 if it enters vertex ''v''<sub>''i''</sub> and 0 otherwise (many authors use the opposite sign convention).
The incidence matrix of a directed graph is a matrix B where n and m are the number of vertices and edges respectively, such that if the edge e<sub>j</sub> leaves vertex v<sub>i</sub>, 1 if it enters vertex v<sub>i</sub> and 0 otherwise (many authors use the opposite sign convention).
有向图的关联矩阵是一个矩阵 b，其中 n 和 m 分别是顶点和边的数目，这样如果边 e < sub > j </sub > 离开顶点 v < sub > i </sub > ，1如果它进入顶点 v < sub > i </sub > 和0，否则(许多作者使用相反的符号约定)。
The ''oriented incidence matrix'' of an undirected graph is the incidence matrix, in the sense of directed graphs, of any [[Orientation (graph theory)|orientation]] of the graph. That is, in the column of edge ''e'', there is one 1 in the row corresponding to one vertex of ''e'' and one −1 in the row corresponding to the other vertex of ''e'', and all other rows have 0. The oriented incidence matrix is unique [[up to]] negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.
The oriented incidence matrix of an undirected graph is the incidence matrix, in the sense of directed graphs, of any orientation of the graph. That is, in the column of edge e, there is one 1 in the row corresponding to one vertex of e and one −1 in the row corresponding to the other vertex of e, and all other rows have 0. The oriented incidence matrix is unique up to negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.
无向图的有向关联矩阵是无向图的有向关联矩阵，在有向图的意义上是无向图的任何方向。也就是说，在 e 边的列中，对应 e 的一个顶点的行中有一个1，对应 e 的另一个顶点的行中有一个 -1，而所有其他行都有0。方向关联矩阵是唯一的，直到否定任何列，因为否定列的条目相当于反转边的方向。
The unoriented incidence matrix of a graph ''G'' is related to the [[adjacency matrix]] of its [[line graph]] ''L''(''G'') by the following theorem:
The unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L(G) by the following theorem:
图 g 的无向关联矩阵与其线图 l (g)的邻接矩阵有以下定理关系:
: <math>A(L(G)) = B(G)^\textsf{T}B(G) - 2I_m.</math>
<math>A(L(G)) = B(G)^\textsf{T}B(G) - 2I_m.</math>
A (l (g)) = b (g) ^ textsf { t } b (g)-2I _ m. </math >
where ''A''(''L''(''G'')) is the adjacency matrix of the line graph of ''G'', ''B''(''G'') is the incidence matrix, and I<sub>''m''</sub> is the [[identity matrix]] of dimension ''m''.
where A(L(G)) is the adjacency matrix of the line graph of G, B(G) is the incidence matrix, and I<sub>m</sub> is the identity matrix of dimension m.
其中 a (l (g))是 g 的线图的邻接矩阵，b (g)是关联矩阵，i < sub > m </sub > 是维数 m 的恒等矩阵。
The discrete [[Kirchhoff matrix|Laplacian]] (or Kirchhoff matrix) is obtained from the oriented incidence matrix ''B''(''G'') by the formula
The discrete Laplacian (or Kirchhoff matrix) is obtained from the oriented incidence matrix B(G) by the formula
离散的拉普拉斯矩阵(或基尔霍夫矩阵)是由定向的关联矩阵 b (g)通过公式得到的
: <math>B(G) B(G)^\textsf{T}.</math>
<math>B(G) B(G)^\textsf{T}.</math>
B (g) b (g) ^ textsf { t }
The integral [[cycle space]] of a graph is equal to the [[null space]] of its oriented incidence matrix, viewed as a matrix over the [[integers]] or [[Real numbers|real]] or [[complex numbers]]. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element [[Field (mathematics)|field]].
The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.
图的整圈空间等于其有向关联矩阵的零空间，可以看作是整数或实数或复数上的矩阵。二元循环空间是有向或无向关联矩阵的零空间，可以看作是二元场上的矩阵。
===Signed and bidirected graphs===
The incidence matrix of a [[signed graph]] is a generalization of the oriented incidence matrix. It is the incidence matrix of any [[bidirected graph]] that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph. The column of a negative edge has either a 1 or a −1 in both rows. The line graph and Kirchhoff matrix properties generalize to signed graphs.
The incidence matrix of a signed graph is a generalization of the oriented incidence matrix. It is the incidence matrix of any bidirected graph that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph. The column of a negative edge has either a 1 or a −1 in both rows. The line graph and Kirchhoff matrix properties generalize to signed graphs.
有符号图的关联矩阵是有向关联矩阵的推广。它是定向给定有符号图的任意双向图的关联矩阵。正边的列在行中对应一个端点有一个1，在行中对应另一个端点有一个 -1，就像普通(无符号)图中的边一样。负边的列在两行中都有1或 -1。线图和 Kirchhoff 矩阵性质推广到符号图。
===Multigraphs===
The definitions of incidence matrix apply to graphs with [[loop (graph theory)|loops]] and [[multiple edges]]. The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.
The definitions of incidence matrix apply to graphs with loops and multiple edges. The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.
关联矩阵的定义适用于有圈和多边的图。对应于一个循环的有向关联矩阵的列都是零，除非这个图是带符号的，而且循环是负的; 那么这个列除了其关联顶点行的 ± 2之外都是零。
===Hypergraphs===
Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a [[hypergraph]] can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a hypergraph can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
因为普通图的边只能有两个顶点(每个顶点一个) ，所以图的关联矩阵列只能有两个非零条目。相比之下，超图可以将多个顶点分配给一条边; 因此，一般的非负整数矩阵描述超图。
==Incidence structures==
The ''incidence matrix'' of an [[incidence structure]] ''C'' is a {{nowrap|''p'' × ''q''}} matrix ''B'' (or its transpose), where ''p'' and ''q'' are the number of ''points'' and ''lines'' respectively, such that {{nowrap|1=''B''<sub>''i'',''j''</sub> = 1}} if the point ''p''<sub>i</sub> and line ''L''<sub>''j''</sub> are incident and 0 otherwise. In this case, the incidence matrix is also a [[biadjacency matrix]] of the [[Levi graph]] of the structure. As there is a [[hypergraph]] for every Levi graph, and ''vice versa'', the incidence matrix of an incidence structure describes a hypergraph.
The incidence matrix of an incidence structure C is a matrix B (or its transpose), where p and q are the number of points and lines respectively, such that if the point p<sub>i</sub> and line L<sub>j</sub> are incident and 0 otherwise. In this case, the incidence matrix is also a biadjacency matrix of the Levi graph of the structure. As there is a hypergraph for every Levi graph, and vice versa, the incidence matrix of an incidence structure describes a hypergraph.
关联结构 c 的关联矩阵是一个矩阵 b (或其转置) ，其中 p 和 q 分别是点和线的数目，如果点 p < sub > i </sub > 和线 l < sub > j </sub > 是关联的，否则为0。在这种情况下，关联矩阵也是 Levi 图的双邻接矩阵的结构。由于每个 Levi 图都有一个超图，反之亦然，关联结构的关联矩阵描述了一个超图。
===Finite geometries===
An important example is a [[finite geometry]]. For instance, in a finite plane, ''X'' is the set of points and ''Y'' is the set of lines. In a finite geometry of higher dimension, ''X'' could be the set of points and ''Y'' could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, ''X'' could be the set of all subspaces of one dimension ''d'' and ''Y'' the set of all subspaces of another dimension ''e'', with incidence defined as containment.
An important example is a finite geometry. For instance, in a finite plane, X is the set of points and Y is the set of lines. In a finite geometry of higher dimension, X could be the set of points and Y could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, X could be the set of all subspaces of one dimension d and Y the set of all subspaces of another dimension e, with incidence defined as containment.
一个重要的例子是有限几何。例如，在有限平面中，x 是点的集合，y 是线的集合。在高维有限几何中，x 可以是点的集合，y 可以是低于整个空间维数的一维子空间(超平面)的集合; 或者，更一般地，x 可以是一维子空间 d 的所有子空间的集合，y 可以是另一维子空间 e 的所有子空间的集合，其关联度定义为包含。
===Polytopes===
In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.<ref>{{citation|first=H.S.M.|last=Coxeter|author-link=Coxeter|title=[[Regular Polytopes (book)|Regular Polytopes]]|year=1973|edition=3rd|origyear=1963|publisher=Dover|isbn=0-486-61480-8|pages=[https://archive.org/details/regularpolytopes0000coxe/page/166 166-167]}}</ref>
In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.
类似地，多胞体中尺寸差一个的细胞之间的关系可以用关联矩阵表示。
===Block designs===
Another example is a [[block design]]. Here ''X'' is a finite set of "points" and ''Y'' is a class of subsets of ''X'', called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove [[Fisher's inequality]], a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points.<ref>{{citation|page=99|first=Herbert John|last=Ryser|title=Combinatorial Mathematics|series=The Carus Mathematical Monographs #14|publisher=The Mathematical Association of America|year=1963}}</ref> Considering the blocks as a system of sets, the [[Permanent (mathematics)|permanent]] of the incidence matrix is the number of [[system of distinct representatives|systems of distinct representatives]] (SDRs).
Another example is a block design. Here X is a finite set of "points" and Y is a class of subsets of X, called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove Fisher's inequality, a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points. Considering the blocks as a system of sets, the permanent of the incidence matrix is the number of systems of distinct representatives (SDRs).
另一个例子是块设计。这里 x 是一个有限的“点”集合，而 y 是 x 的一类子集，称为“块” ，受依赖于设计类型的规则的制约。关联矩阵是块设计理论中的一个重要工具。例如，它可以用来证明 Fisher 不等式，一个平衡不完全2- 设计(bibd)的基本定理，块的数目至少是点的数目。将区组看作一个集合系统，关联矩阵的常数是不同代表系统的个数。
==References==
{{reflist}}
==Further reading==
* {{citation | last=Diestel | first=Reinhard | title=Graph Theory | series=[[Graduate Texts in Mathematics]] | volume=173 | edition=3rd | date=2005 | publisher=Springer-Verlag | isbn=3-540-26183-4}}
* Jonathan L Gross, Jay Yellen, ''Graph Theory and its applications'', second edition, 2006 (p 97, Incidence Matrices for undirected graphs; p 98, incidence matrices for digraphs)
==External links==
{{Commons category|Incidence matrices of graphs}}
{{Wiktionary}}
* {{mathworld | urlname = IncidenceMatrix | title = Incidence matrix}}
{{Graph representations}}
[[Category:Algebraic graph theory]]
Category:Algebraic graph theory
范畴: 代数图论
[[Category:Combinatorics]]
Category:Combinatorics
分类: 组合数学
[[Category:Matrices]]
Category:Matrices
分类: 矩阵
[[Category:Graph data structures]]
Category:Graph data structures
类别: 图形数据结构
<noinclude>
<small>This page was moved from [[wikipedia:en:Incidence matrix]]. Its edit history can be viewed at [[关联矩阵/edithistory]]</small></noinclude>
[[Category:待整理页面]]
In [[mathematics]], an '''incidence matrix''' is a [[Matrix (mathematics)|matrix]] that shows the relationship between two classes of objects. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element of ''X'' and one column for each element of ''Y''. The entry in row ''x'' and column ''y'' is 1 if ''x'' and ''y'' are related (called ''incident'' in this context) and 0 if they are not. There are variations; see below.
In mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 if x and y are related (called incident in this context) and 0 if they are not. There are variations; see below.
在数学中，关联矩阵是表示两类对象之间关系的矩阵。如果第一个类是 x，第二个类是 y，那么这个矩阵对于 x 的每个元素有一行，对于 y 的每个元素有一列。如果 x 和 y 是相关的(在本文中称为 incident) ，则行 x 和列 y 中的条目为1，如果它们不是相关的，则为0。这里有一些变化，请看下面。
==Graph theory==
Incidence matrices are frequently used in [[graph theory]].
Incidence matrices are frequently used in graph theory.
关联矩阵是图论中经常使用的一种方法。
===Undirected and directed graphs===
[[Image:Labeled undirected graph.svg|thumb|250px|An undirected graph.]]
An undirected graph.
无向图一个无向图。
In graph theory an [[undirected graph]] has two kinds of incidence matrices: unoriented and oriented.
In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented.
在图论中，无向图有两种关联矩阵: 无向图和有向图。
The ''unoriented incidence matrix'' (or simply ''incidence matrix'') of an undirected graph is a {{nowrap|''n'' × ''m''}} [[Matrix (math)|matrix]] ''B'', where ''n'' and ''m'' are the numbers of [[Vertex (graph theory)|vertices]] and [[Edge (graph theory)|edge]]s respectively, such that {{nowrap|1=''B''<sub>''i'',''j''</sub> = 1}} if the vertex ''v''<sub>''i''</sub> and edge ''e''<sub>''j''</sub> are incident and 0 otherwise.
The unoriented incidence matrix (or simply incidence matrix) of an undirected graph is a matrix B, where n and m are the numbers of vertices and edges respectively, such that if the vertex v<sub>i</sub> and edge e<sub>j</sub> are incident and 0 otherwise.
无向图的无向关联矩阵是一个矩阵 b，其中 n 和 m 分别是顶点和边的数目，如果顶点 v < sub > i </sub > 和边 e < sub > j </sub > 是关联的，否则为0。
For example the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1–4) and 4 columns (corresponding to the four edges, e1–e4):
For example the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1–4) and 4 columns (corresponding to the four edges, e1–e4):
例如，右边所示的无向图的关联矩阵是一个由4行(对应4个顶点，1-4)和4列(对应4条边，e1-e4)组成的矩阵:
{|
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{| align=left class=wikitable
{| align=left class=wikitable
{ | align = left class = wikable
|-
|-
|-
! !! e<sub>1</sub> !! e<sub>2</sub> !! e<sub>3</sub> !! e<sub>4</sub>
! !! e<sub>1</sub> !! e<sub>2</sub> !! e<sub>3</sub> !! e<sub>4</sub>
!!!电子邮件！2! ！3 </sub > ！E < sub > 4 </sub >
|-
|-
|-
!1
!1
!1
|1||1||1||0
|1||1||1||0
|1||1||1||0
|-
|-
|-
!2
!2
!2
|1||0||0||0
|1||0||0||0
|1||0||0||0
|-
|-
|-
!3
!3
!3
|0||1||0||1
|0||1||0||1
|0||1||0||1
|-
|-
|-
!4
!4
!4
|0||0||1||1
|0||0||1||1
|0||0||1||1
|}
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<math>
<math>
《数学》
\begin{pmatrix}
\begin{pmatrix}
开始{ pmatrix }
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
\end{pmatrix}.
\end{pmatrix}.
结束{ pmatrix }。
</math>
</math>
数学
|}
|}
|}
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.
如果我们看一下关联矩阵，我们会发现每一列的总和等于2。这是因为每条边都有一个顶点连接到每个端点。
The ''incidence matrix'' of a [[directed graph]] is a {{nowrap|''n'' × ''m''}} matrix ''B'' where ''n'' and ''m'' are the number of vertices and edges respectively, such that {{nowrap|1=''B''<sub>''i'',''j''</sub> = −1}} if the edge ''e''<sub>''j''</sub> leaves vertex ''v''<sub>i</sub>, 1 if it enters vertex ''v''<sub>''i''</sub> and 0 otherwise (many authors use the opposite sign convention).
The incidence matrix of a directed graph is a matrix B where n and m are the number of vertices and edges respectively, such that if the edge e<sub>j</sub> leaves vertex v<sub>i</sub>, 1 if it enters vertex v<sub>i</sub> and 0 otherwise (many authors use the opposite sign convention).
有向图的关联矩阵是一个矩阵 b，其中 n 和 m 分别是顶点和边的数目，这样如果边 e < sub > j </sub > 离开顶点 v < sub > i </sub > ，1如果它进入顶点 v < sub > i </sub > 和0，否则(许多作者使用相反的符号约定)。
The ''oriented incidence matrix'' of an undirected graph is the incidence matrix, in the sense of directed graphs, of any [[Orientation (graph theory)|orientation]] of the graph. That is, in the column of edge ''e'', there is one 1 in the row corresponding to one vertex of ''e'' and one −1 in the row corresponding to the other vertex of ''e'', and all other rows have 0. The oriented incidence matrix is unique [[up to]] negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.
The oriented incidence matrix of an undirected graph is the incidence matrix, in the sense of directed graphs, of any orientation of the graph. That is, in the column of edge e, there is one 1 in the row corresponding to one vertex of e and one −1 in the row corresponding to the other vertex of e, and all other rows have 0. The oriented incidence matrix is unique up to negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.
无向图的有向关联矩阵是无向图的有向关联矩阵，在有向图的意义上是无向图的任何方向。也就是说，在 e 边的列中，对应 e 的一个顶点的行中有一个1，对应 e 的另一个顶点的行中有一个 -1，而所有其他行都有0。方向关联矩阵是唯一的，直到否定任何列，因为否定列的条目相当于反转边的方向。
The unoriented incidence matrix of a graph ''G'' is related to the [[adjacency matrix]] of its [[line graph]] ''L''(''G'') by the following theorem:
The unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L(G) by the following theorem:
图 g 的无向关联矩阵与其线图 l (g)的邻接矩阵有以下定理关系:
: <math>A(L(G)) = B(G)^\textsf{T}B(G) - 2I_m.</math>
<math>A(L(G)) = B(G)^\textsf{T}B(G) - 2I_m.</math>
A (l (g)) = b (g) ^ textsf { t } b (g)-2I _ m. </math >
where ''A''(''L''(''G'')) is the adjacency matrix of the line graph of ''G'', ''B''(''G'') is the incidence matrix, and I<sub>''m''</sub> is the [[identity matrix]] of dimension ''m''.
where A(L(G)) is the adjacency matrix of the line graph of G, B(G) is the incidence matrix, and I<sub>m</sub> is the identity matrix of dimension m.
其中 a (l (g))是 g 的线图的邻接矩阵，b (g)是关联矩阵，i < sub > m </sub > 是维数 m 的恒等矩阵。
The discrete [[Kirchhoff matrix|Laplacian]] (or Kirchhoff matrix) is obtained from the oriented incidence matrix ''B''(''G'') by the formula
The discrete Laplacian (or Kirchhoff matrix) is obtained from the oriented incidence matrix B(G) by the formula
离散的拉普拉斯矩阵(或基尔霍夫矩阵)是由定向的关联矩阵 b (g)通过公式得到的
: <math>B(G) B(G)^\textsf{T}.</math>
<math>B(G) B(G)^\textsf{T}.</math>
B (g) b (g) ^ textsf { t }
The integral [[cycle space]] of a graph is equal to the [[null space]] of its oriented incidence matrix, viewed as a matrix over the [[integers]] or [[Real numbers|real]] or [[complex numbers]]. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element [[Field (mathematics)|field]].
The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.
图的整圈空间等于其有向关联矩阵的零空间，可以看作是整数或实数或复数上的矩阵。二元循环空间是有向或无向关联矩阵的零空间，可以看作是二元场上的矩阵。
===Signed and bidirected graphs===
The incidence matrix of a [[signed graph]] is a generalization of the oriented incidence matrix. It is the incidence matrix of any [[bidirected graph]] that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph. The column of a negative edge has either a 1 or a −1 in both rows. The line graph and Kirchhoff matrix properties generalize to signed graphs.
The incidence matrix of a signed graph is a generalization of the oriented incidence matrix. It is the incidence matrix of any bidirected graph that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph. The column of a negative edge has either a 1 or a −1 in both rows. The line graph and Kirchhoff matrix properties generalize to signed graphs.
有符号图的关联矩阵是有向关联矩阵的推广。它是定向给定有符号图的任意双向图的关联矩阵。正边的列在行中对应一个端点有一个1，在行中对应另一个端点有一个 -1，就像普通(无符号)图中的边一样。负边的列在两行中都有1或 -1。线图和 Kirchhoff 矩阵性质推广到符号图。
===Multigraphs===
The definitions of incidence matrix apply to graphs with [[loop (graph theory)|loops]] and [[multiple edges]]. The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.
The definitions of incidence matrix apply to graphs with loops and multiple edges. The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.
关联矩阵的定义适用于有圈和多边的图。对应于一个循环的有向关联矩阵的列都是零，除非这个图是带符号的，而且循环是负的; 那么这个列除了其关联顶点行的 ± 2之外都是零。
===Hypergraphs===
Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a [[hypergraph]] can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a hypergraph can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
因为普通图的边只能有两个顶点(每个顶点一个) ，所以图的关联矩阵列只能有两个非零条目。相比之下，超图可以将多个顶点分配给一条边; 因此，一般的非负整数矩阵描述超图。
==Incidence structures==
The ''incidence matrix'' of an [[incidence structure]] ''C'' is a {{nowrap|''p'' × ''q''}} matrix ''B'' (or its transpose), where ''p'' and ''q'' are the number of ''points'' and ''lines'' respectively, such that {{nowrap|1=''B''<sub>''i'',''j''</sub> = 1}} if the point ''p''<sub>i</sub> and line ''L''<sub>''j''</sub> are incident and 0 otherwise. In this case, the incidence matrix is also a [[biadjacency matrix]] of the [[Levi graph]] of the structure. As there is a [[hypergraph]] for every Levi graph, and ''vice versa'', the incidence matrix of an incidence structure describes a hypergraph.
The incidence matrix of an incidence structure C is a matrix B (or its transpose), where p and q are the number of points and lines respectively, such that if the point p<sub>i</sub> and line L<sub>j</sub> are incident and 0 otherwise. In this case, the incidence matrix is also a biadjacency matrix of the Levi graph of the structure. As there is a hypergraph for every Levi graph, and vice versa, the incidence matrix of an incidence structure describes a hypergraph.
关联结构 c 的关联矩阵是一个矩阵 b (或其转置) ，其中 p 和 q 分别是点和线的数目，如果点 p < sub > i </sub > 和线 l < sub > j </sub > 是关联的，否则为0。在这种情况下，关联矩阵也是 Levi 图的双邻接矩阵的结构。由于每个 Levi 图都有一个超图，反之亦然，关联结构的关联矩阵描述了一个超图。
===Finite geometries===
An important example is a [[finite geometry]]. For instance, in a finite plane, ''X'' is the set of points and ''Y'' is the set of lines. In a finite geometry of higher dimension, ''X'' could be the set of points and ''Y'' could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, ''X'' could be the set of all subspaces of one dimension ''d'' and ''Y'' the set of all subspaces of another dimension ''e'', with incidence defined as containment.
An important example is a finite geometry. For instance, in a finite plane, X is the set of points and Y is the set of lines. In a finite geometry of higher dimension, X could be the set of points and Y could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, X could be the set of all subspaces of one dimension d and Y the set of all subspaces of another dimension e, with incidence defined as containment.
一个重要的例子是有限几何。例如，在有限平面中，x 是点的集合，y 是线的集合。在高维有限几何中，x 可以是点的集合，y 可以是低于整个空间维数的一维子空间(超平面)的集合; 或者，更一般地，x 可以是一维子空间 d 的所有子空间的集合，y 可以是另一维子空间 e 的所有子空间的集合，其关联度定义为包含。
===Polytopes===
In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.<ref>{{citation|first=H.S.M.|last=Coxeter|author-link=Coxeter|title=[[Regular Polytopes (book)|Regular Polytopes]]|year=1973|edition=3rd|origyear=1963|publisher=Dover|isbn=0-486-61480-8|pages=[https://archive.org/details/regularpolytopes0000coxe/page/166 166-167]}}</ref>
In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.
类似地，多胞体中尺寸差一个的细胞之间的关系可以用关联矩阵表示。
===Block designs===
Another example is a [[block design]]. Here ''X'' is a finite set of "points" and ''Y'' is a class of subsets of ''X'', called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove [[Fisher's inequality]], a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points.<ref>{{citation|page=99|first=Herbert John|last=Ryser|title=Combinatorial Mathematics|series=The Carus Mathematical Monographs #14|publisher=The Mathematical Association of America|year=1963}}</ref> Considering the blocks as a system of sets, the [[Permanent (mathematics)|permanent]] of the incidence matrix is the number of [[system of distinct representatives|systems of distinct representatives]] (SDRs).
Another example is a block design. Here X is a finite set of "points" and Y is a class of subsets of X, called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove Fisher's inequality, a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points. Considering the blocks as a system of sets, the permanent of the incidence matrix is the number of systems of distinct representatives (SDRs).
另一个例子是块设计。这里 x 是一个有限的“点”集合，而 y 是 x 的一类子集，称为“块” ，受依赖于设计类型的规则的制约。关联矩阵是块设计理论中的一个重要工具。例如，它可以用来证明 Fisher 不等式，一个平衡不完全2- 设计(bibd)的基本定理，块的数目至少是点的数目。将区组看作一个集合系统，关联矩阵的常数是不同代表系统的个数。
==References==
{{reflist}}
==Further reading==
* {{citation | last=Diestel | first=Reinhard | title=Graph Theory | series=[[Graduate Texts in Mathematics]] | volume=173 | edition=3rd | date=2005 | publisher=Springer-Verlag | isbn=3-540-26183-4}}
* Jonathan L Gross, Jay Yellen, ''Graph Theory and its applications'', second edition, 2006 (p 97, Incidence Matrices for undirected graphs; p 98, incidence matrices for digraphs)
==External links==
{{Commons category|Incidence matrices of graphs}}
{{Wiktionary}}
* {{mathworld | urlname = IncidenceMatrix | title = Incidence matrix}}
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