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# IncidenceMatrix

 IncidenceMatrix[g] gives the vertex-edge incidence matrix of the graph g.
• For an undirected graph, an entry of the incidence matrix is given by:
 0 vertex is not incident to edge 1 vertex is incident to edge 2 vertex is incident to edge and a self-loop
• For a directed graph, an entry of the incidence matrix is given by:
 0 vertex is not incident to edge 1 vertex is incident to edge and is the head of -1 vertex is incident to edge and is the tail of 2 vertex is incident to edge and a self-loop
• The vertices are assumed to be in the order given by VertexList[g] and the edges are assumed to be in the order given by EdgeList[g].
• The incidence matrix for a graph will have an × matrix, where is the number of vertices and is the number of edges, counting multiplicity.
The incidence matrix of an undirected graph:
The incidence matrix of a directed graph:
The incidence matrix of an undirected graph:
 Out[1]=
 Out[2]//MatrixForm=

The incidence matrix of a directed graph:
 Out[1]=
 Out[2]//MatrixForm=
 Scope   (4)
The incidence matrix of an undirected graph has no negative entries:
The sum of the entries in any column is 2:
The incidence matrix of a directed graph has some negative entries:
If there are no self-loops, the sum of the entries in any column is 0:
The incidence matrix of a graph with self-loops has some entries equal to 2:
IncidenceMatrix works with large graphs:
Use MatrixPlot to visualize the matrix:
Rows and columns of the incidence matrix correspond to VertexList and EdgeList:
The number of rows of the incidence matrix is equal to the number of vertices:
The number of columns is equal to the number of edges:
Use IncidenceMatrix to construct a graph from an incidence matrix:
The adjacency matrix of a line graph can be computed by its IncidenceMatrix:
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