Seidel adjacency matrix

In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices. It is also called the Seidel matrix or—its original name—the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the adjacency matrix of G from the adjacency matrix of the complement of G.

The multiset of eigenvalues of this matrix is called the Seidel spectrum.

The Seidel matrix was introduced by J. H. van Lint and Johan Jacob Seidel [de; nl] in 1966 and extensively exploited by Seidel and coauthors.

The Seidel matrix of G is also the adjacency matrix of a signed complete graph K_G in which the edges of G are negative and the edges not in G are positive. It is also the adjacency matrix of the two-graph associated with G and K_G.

The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graphs.

References

van Lint, J. H., and Seidel, J. J. (1966), Equilateral point sets in elliptic geometry. Indagationes Mathematicae, vol. 28 (= Proc. Kon. Ned. Aka. Wet. Ser. A, vol. 69), pp. 335–348.
Seidel, J. J. (1976), A survey of two-graphs. In: Colloquio Internazionale sulle Teorie Combinatorie (Proceedings, Rome, 1973), vol. I, pp. 481–511. Atti dei Convegni Lincei, No. 17. Accademia Nazionale dei Lincei, Rome.
Seidel, J. J. (1991), ed. D.G. Corneil and R. Mathon, Geometry and Combinatorics: Selected Works of J. J. Seidel. Boston: Academic Press. Many of the articles involve the Seidel matrix.
Seidel, J. J. (1968), Strongly Regular Graphs with (−1,1,0) Adjacency Matrix Having Eigenvalue 3. Linear Algebra and its Applications 1, 281–298.

This graph theory-related article is a stub. You can help Wikipedia by expanding it.

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