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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。<br>

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。<br>

本词条由信白初步翻译<br>

本词条由信白初步翻译<br>

==Undirected and directed graphs==

==Undirected and directed graphs==

'''<font color="#ff8000">无向和有向图 Undirected And Directed Graphs</font>'''

'''<font color="#ff8000">无向图和有向图 Undirected And Directed Graphs</font>'''

[[Image:Labeled undirected graph.svg|thumb|250px|An undirected graph.]]

[[Image:Labeled undirected graph.svg|thumb|250px|An undirected graph.]]

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，'''<font color="#ff8000">无向图 Undirected Graph</font>'''有两种关联矩阵: 无向矩阵和有向关联矩阵。

在图论中，'''<font color="#ff8000">无向图 Undirected Graph</font>'''有两种关联矩阵: 无向关联矩阵和有向关联矩阵。

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.

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。

无向图的无向关联矩阵（也称为简单的关联矩阵）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，则结果为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):

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：

如图右所示的无向图，它的关联矩阵是一个4行(分别对应4个顶点，1-4)4列(各自对应4条边，e₁-e₄)组成的矩阵：

{|

{|

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.

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.

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.

'''<font color="#ff8000">有符号图 Signed Graph</font>'''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'''<font color="#ff8000">(无符号)图 Unsigned Graph</font>'''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。

'''<font color="#ff8000">有符号图 Signed Graph</font>'''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，并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'''<font color="#ff8000">(无符号)图 Unsigned Graph</font>'''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==

==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'''<font color="#ff8000">多重关联图 Multigraphs</font>'''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。

'''<font color="#ff8000">多重关联图 Multigraphs</font>'''的定义：多重关联图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，否则该列除其入射顶点行中的±2外均为零。

==Hypergraphs==

==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.

==Incidence structures==

==Incidence structures==

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.

举一个有限几何的重要例子。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，关联度定义也包含在内。

举一个有限几何的重要例子。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'' 仍可以是点的集合，''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，关联度定义也包含在内。

==Polytopes==

==Polytopes==

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).

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'''<font color="#ff8000">Fisher不等式 Fisher's inequality</font>'''，一个平衡不完全2- 设计(BIBDs)的基本定理，块的数目至少是点的数目。将块看作一个集合系统，关联矩阵的常数是不同代表系统的个数（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” ，它受设计类型规则的制约。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'''<font color="#ff8000">Fisher不等式 Fisher's inequality</font>'''，一个平衡不完全2- 设计(BIBDs)的基本定理，块的数目至少是点的数目。将块看作一个集合系统，关联矩阵的常数是不同代表系统的个数（SDRs）。

==References==

==References==