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, 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

REFERENCES

Bruck, R. H. and Ryser, H. J. "The Nonexistence of Certain Finite Projective Planes." Canad. J. Math. 1, 88-93, 1949.

Kirchhoff, G. "Über die Auflösung der Gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer Ströme geführt wird." Ann. Phys. Chem. 72, 497-508, 1847.

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 135-136, 1990.