Incidence Matrix

The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and iff vertex is incident upon edge (Skiena 1990, p. 135). However, some authors define the incidence matrix to be the transpose of this (including the standard form of the embedding-encoding generalization known as the rigidity matrix), with a column for each vertex and a row for each edge. The physicist Kirchhoff (1847) was the first to define the incidence matrix.

The incidence matrix of a graph (using the first definition) can be computed in the Wolfram Language using IncidenceMatrix[g]. Precomputed incidence matrices for a many named graphs are given in the Wolfram Language by GraphData[graph, "IncidenceMatrix"].

The incidence matrix of a graph and adjacency matrix of its line graph are related by

(1)

where is the identity matrix (Skiena 1990, p. 136).

For a -D polytope , the incidence matrix is defined by

(2)

The th row shows which s surround , and the th column shows which s bound . Incidence matrices are also used to specify projective planes. The incidence matrices for a tetrahedron are

	1
1	1	1	1	1

	1	0	0	0	1	1
	0	1	0	1	0	1
	0	0	1	1	1	0
	1	1	1	0	0	0

	0	1	1	0
	1	0	1	0
	1	1	0	0
	1	0	0	1
	0	1	0	1
	0	0	1	1

	1
	1
	1
	1