Incidence matrix

In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each mapping from X to Y. The entry in row x and column y is 1 if the vertex x is part of (called incident in this context) the mapping that corresponds to y, and 0 if it is not. There are variations; see below.

Graph theory

Incidence matrix is a common graph representation in graph theory. It is different to an adjacency matrix, which encodes the relation of vertex-vertex pairs.

Undirected and directed graphs

In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented.

The unoriented incidence matrix (or simply incidence matrix) of an undirected graph is a $n\times m$ matrix B, where n and m are the numbers of vertices and edges respectively, such that

B_{ij}=\left\{{\begin{array}{rl}\,1&{\text{if vertex }}v_{i}{\text{ is incident with edge }}e_{j},\\0&{\text{otherwise.}}\end{array}}\right.

For example, the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1–4) and 4 columns (corresponding to the four edges, $e_{1},e_{2},e_{3},e_{4}$ ):

	e₁	e₂	e₃	e₄
1	1	1	1	0
2	1	0	0	0
3	0	1	0	1
4	0	0	1	1

=

${\begin{bmatrix}1&1&1&0\\1&0&0&0\\0&1&0&1\\0&0&1&1\\\end{bmatrix}}.$

If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.

The incidence matrix of a directed graph is a $n\times m$ matrix B where n and m are the number of vertices and edges respectively, such that

B_{ij}=\left\{{\begin{array}{rl}{-1}&{\text{if edge }}e_{j}{\text{ leaves vertex }}v_{i},\\{\phantom {-}}1&{\text{if edge }}e_{j}{\text{ enters vertex }}v_{i},\\{\phantom {-}}0&{\text{otherwise.}}\end{array}}\right.

(Many authors use the opposite sign convention.)

The oriented incidence matrix of an undirected graph is the incidence matrix, in the sense of directed graphs, of any orientation of the graph. That is, in the column of edge e, there is one 1 in the row corresponding to one vertex of e and one −1 in the row corresponding to the other vertex of e, and all other rows have 0. The oriented incidence matrix is unique up to negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.

The unoriented incidence matrix of a graph G is related to the adjacency matrix of its line graph L(G) by the following theorem:

A(L(G))=B(G)^{\textsf {T}}B(G)-2I_{m}.

where A(L(G)) is the adjacency matrix of the line graph of G, B(G) is the incidence matrix, and I_m is the identity matrix of dimension m.

The discrete Laplacian (or Kirchhoff matrix) is obtained from the oriented incidence matrix B(G) by the formula

B(G)B(G)^{\textsf {T}}.

The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element field.

Signed and bidirected graphs

The incidence matrix of a signed graph is a generalization of the oriented incidence matrix. It is the incidence matrix of any bidirected graph that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a −1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph. The column of a negative edge has either a 1 or a −1 in both rows. The line graph and Kirchhoff matrix properties generalize to signed graphs.

Multigraphs

The definitions of incidence matrix apply to graphs with loops and multiple edges. The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.

Weighted graphs

A weighted graph can be represented using the weight of the edge in place of a 1. For example, the incidence matrix of the graph to the right is:

	e₁	e₂	e₃	e₄
1	2	1	5	0
2	2	0	0	0
3	0	1	0	6
4	0	0	5	6

=

${\begin{bmatrix}2&1&5&0\\2&0&0&0\\0&1&0&6\\0&0&5&6\\\end{bmatrix}}.$

Hypergraphs

Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a hypergraph can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.

Incidence structures

The incidence matrix of an incidence structure C is a p × q matrix B (or its transpose), where p and q are the number of points and lines respectively, such that B_i,j = 1 if the point p_i and line L_j are incident and 0 otherwise. In this case, the incidence matrix is also a biadjacency matrix of the Levi graph of the structure. As there is a hypergraph for every Levi graph, and vice versa, the incidence matrix of an incidence structure describes a hypergraph.

Finite geometries

An important example is a finite geometry. For instance, in a finite plane, X is the set of points and Y is the set of lines. In a finite geometry of higher dimension, X could be the set of points and Y could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally, X could be the set of all subspaces of one dimension d and Y the set of all subspaces of another dimension e, with incidence defined as containment.

Polytopes

In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.^[1]

Block designs

Another example is a block design. Here X is a finite set of "points" and Y is a class of subsets of X, called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove Fisher's inequality, a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points.^[2] Considering the blocks as a system of sets, the permanent of the incidence matrix is the number of systems of distinct representatives (SDRs).

References

^ Coxeter, H.S.M. (1973) [1963], Regular Polytopes (3rd ed.), Dover, pp. 166-167, ISBN 0-486-61480-8
^ Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs #14, The Mathematical Association of America, p. 99

External links

Weisstein, Eric W. "Incidence matrix". MathWorld.

[1] Coxeter, H.S.M. (1973) [1963], Regular Polytopes (3rd ed.), Dover, pp. 166-167, ISBN 0-486-61480-8

[2] Ryser, Herbert John (1963), Combinatorial Mathematics, The Carus Mathematical Monographs #14, The Mathematical Association of America, p. 99

[1]

[2]

v t e Graph representations
Data structures	Graph (abstract data type) Adjacency list Edge list Adjacency matrix Incidence matrix
XML-based formats	DGML DotML GEXF GraphML GXL XGMML
Text-based formats	DOT Graph Modelling Language (GML) LCF notation for cubic Hamiltonian graphs Newick format for trees Trivial Graph Format
Related concepts	Graph database Graph drawing Linked data

v t e Incidence structures
Representation	Incidence matrix Incidence graph
Fields	Combinatorics Block design Steiner system Geometry Incidence Projective plane Graph theory Hypergraph Statistics Blocking
Configurations	Complete quadrangle Fano plane Möbius–Kantor configuration Pappus configuration Hesse configuration Desargues configuration Reye configuration Pasch configuration Schläfli double six Cremona–Richmond configuration Kummer configuration Grünbaum–Rigby configuration Klein configuration Dual
Theorems	Sylvester–Gallai theorem De Bruijn–Erdős theorem Szemerédi–Trotter theorem Beck's theorem Bruck–Ryser–Chowla theorem
Applications	Design of experiments Kirkman's schoolgirl problem

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices