Tutte matrix

In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once.

If the set of vertices V has n elements then the Tutte matrix is an n × n matrix A with entries

A_{ij}={\begin{cases}x_{ij}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i<j\\-x_{ji}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i>j\\0\;\;\;\;{\mbox{otherwise}}\end{cases}}

where the x_ij are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables x_ij, i < j ): this coincides with the square of the pfaffian of the matrix A and is non-zero (as a polynomial) if and only if a perfect matching exists. (This polynomial is not the Tutte polynomial of G.)

The Tutte matrix is named after W. T. Tutte, and is a generalisation of the Edmonds matrix for a balanced bipartite graph.

References

R. Motwani, P. Raghavan (1995). Randomized Algorithms. Cambridge University Press. p. 167.
Allen B. Tucker (2004). Computer Science Handbook. CRC Press. p. 12.19. ISBN 1-58488-360-X.
W.T. Tutte (April 1947). "The factorization of linear graphs" (PDF). J. London Math. Soc. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107. Retrieved 2008-06-15.

Matrix classes

Explicitly constrained entries	(0,1) Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Binary 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 Markov Metzler Monomial Moore Nonnegative Partitioned Parisi Pentadiagonal Permutation Persymmetric Polynomial Positive Quaternionic Sign Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Unitary Vandermonde Walsh Z

Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero

Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Diagonalizable Hurwitz Positive-definite Stability Stieltjes

Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Orthonormal Singular Unimodular Unipotent Totally unimodular Weighing

With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Coxeter Derogatory Distance Duplication Elimination Euclidean distance Fundamental (linear differential equation) Generator Gramian Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation Wedderburn X–Y–Z

Used in statistics	Bernoulli Centering Correlation Covariance Dispersion Doubly stochastic Fisher information Hat Precision Stochastic Transition

Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Skew-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 canonical form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Quaternionic matrix Row echelon form Wronskian

List of matrices Category:Matrices

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