BUILT-IN MATHEMATICA SYMBOL

IncidenceMatrix

IncidenceMatrix[g]
gives the vertex-edge incidence matrix of the graph g.

IncidenceMatrix returns a SparseArray object, which can be converted to an ordinary matrix using Normal.

	0	vertex is not incident to edge
	1	vertex is incident to edge
	2	vertex is incident to edge and a self-loop

	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=

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: