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==Signed and bidirected graphs==

==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.

'''<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，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==

'''<font color="#ff8000">多重图 Multigraphs</font>'''

'''<font color="#ff8000">多重图 Multigraphs</font>'''

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 [[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.

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。

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。

===Hypergraphs===

==Hypergraphs==

 

'''<font color="#ff8000">超图 Hypergraphs</font>'''

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.

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==