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
![eta_(ij)^k=<span style=]() {1 if Pi_(k-1)^i belongs to Pi_k^j; 0 if Pi_(k-1)^i does not belong to Pi_k^j. " src="https://mathworld.wolfram.com/images/equations/IncidenceMatrix/NumberedEquation2.svg" style="height:57px; width:272px" /> |
(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
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1 |
0 |
0 |
0 |
1 |
1 |
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0 |
1 |
0 |
1 |
0 |
1 |
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0 |
0 |
1 |
1 |
1 |
0 |
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1 |
1 |
1 |
0 |
0 |
0 |
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0 |
1 |
1 |
0 |
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1 |
0 |
1 |
0 |
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1 |
1 |
0 |
0 |
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1 |
0 |
0 |
1 |
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0 |
1 |
0 |
1 |
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0 |
0 |
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.
