关联矩阵

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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.

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 Incidence Matrix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

图论 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表示方法。

Undirected and directed graphs

无向和有向图 Undirected And Directed Graphs

文件:Labeled undirected graph.svg

An undirected graph.

图：An undirected graph.

一个无向图。

In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented.

在图论中，无向图 Undirected Graph有两种关联矩阵: 无向矩阵和有向关联矩阵。

The unoriented incidence matrix (or simply incidence matrix) of an undirected graph is a n × m matrix B, where n and m are the numbers of vertices and edges respectively, such that B_i,j = 1 if the vertex v_i and edge e_j 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_i and edge e_j are incident and 0 otherwise.

无向图的无向关联矩阵（或者简单的关联矩阵）是一个矩阵B，其中n和m 分别是顶点和边的数目，如果顶点 v_i 和边 e_j是关联的，则为1，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):

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：

{ | align = left class = wikable

	e₁	e₂	e₃	e₄		e₁	e₂	e₃	e₄		电子邮件！2! ！3 ！E < sub > 4
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

= = =

[math]\displaystyle{ \lt math\gt 《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.

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 n × m matrix B where n and m are the number of vertices and edges respectively, such that B_i,j = −1 if the edge e_j leaves vertex v_i, 1 if it enters vertex v_i 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_j leaves vertex v_i, 1 if it enters vertex v_i 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_j 离开顶点 v_i，为1，如果它进入顶点 v_i ，为0(许多作者使用相反的符号约定)。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_j leaves vertex v_i, 1 if it enters vertex v_i and 0 otherwise (many authors use the opposite sign convention).

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.

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在有向图的意义上是图的任何方向的关联矩阵。也就是说，在边e的列中，对应于e的一个顶点的行中有一个1，对应于e的另一个顶点的行中有一个−1，所有其他行都有0。定向关联矩阵是唯一的，直到任何列取反为止，因为对列中的条目求反对应于反转边的方向。

The unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L(G) by the following theorem:

图G的无向关联矩阵与其线图 Line GraphL(G)的邻接矩阵有以下定理关系:

[math]\displaystyle{ A(L(G)) = B(G)^\textsf{T}B(G) - 2I_m. }[/math]

[math]\displaystyle{ 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_m is the identity matrix of dimension m.

其中A（L(G)）是G的线图的邻接矩阵 Adjacency Matrix，B(G)是关联矩阵，I_m 是维数为m的单位矩阵 Identity Matrix。

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) Laplacian (or Kirchhoff Matrix)是由定向的关联矩阵B(G)通过公式得到的

[math]\displaystyle{ B(G) B(G)^\textsf{T}. }[/math]

[math]\displaystyle{ 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 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.

图的圈空间 Cycle Space等j价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，也可以看作是二元场上的矩阵。

Signed and bidirected graphs

有符号双向图 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 Signed Graph的关联矩阵是有向关联矩阵的推广。它是任意双向图的关联矩阵，为给定的有符号图定向。正边的列在对应一个端点的行有1，在对应于另一个端点的行中有 -1，就像普通(无符号)图 Unsigned Graph中的边一样。负边的列在两行中都有1或 -1。线图和 Kirchhoff 矩阵性质都能推广到符号图中。

Multigraphs

多重图 Multigraphs 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.

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.

关于多重关联图 Multigraphs的定义，与循环相对应的定向关联矩阵的列均为零，除非图有符号且循环为负；则该列除其入射顶点行中的±2外均为零。

Hypergraphs

超图 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.

由于一般图的边只能有两个顶点（每端一个），图的关联矩阵列只能有两个非零项。相比之下，超图可以有多个顶点指定给一条边；因此，一般的非负整数矩阵描述了超图。

Incidence structures

The incidence matrix of an incidence structure C is a p × q matrix B (or its transpose), where p and q are the number of points and lines respectively, such that B_i,j = 1 if the point p_i and line L_j 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_i and line L_j 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 和线 l < sub > j 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.

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.^[1]

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.^[2] Considering the blocks as a system of sets, the permanent of the incidence matrix is the number of 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

↑ Coxeter, H.S.M. (1973) [1963], Regular Polytopes (3rd ed.), Dover, pp. 166-167, ISBN 0-486-61480-8
↑ Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs #14, The Mathematical Association of America, p. 99

External links

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分类: 组合数学

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分类: 矩阵

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类别: 图形数据结构

This page was moved from wikipedia:en:Incidence matrix. Its edit history can be viewed at 关联矩阵/edithistory

[1] Coxeter, H.S.M. (1973) [1963], Regular Polytopes (3rd ed.), Dover, pp. 166-167, ISBN 0-486-61480-8

[2] Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs #14, The Mathematical Association of America, p. 99

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