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Seidel adjacency matrix

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Matrix in graph theory (mathematics)

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Inmathematics, ingraph theory, theSeidel adjacency matrix of asimple undirected graphG is asymmetric 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 theSeidel matrix or – its original name – the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting theadjacency matrix ofG from the adjacency matrix of thecomplement ofG.

Themultiset ofeigenvalues of this matrix is called theSeidel spectrum.

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

The Seidel matrix ofG is also the adjacency matrix of asigned complete graphK_G in which the edges ofG are negative and the edges not inG are positive. It is also the adjacency matrix of thetwo-graph associated withG andK_G.

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

References

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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.

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 oneigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions onproducts orinverses	Congruent Idempotent orProjection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
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Used instatistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used ingraph 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
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