Seidel adjacency matrix

In mathematics, in graph theory, the Seidel adjacency matrix of a simple graph G (also called the Seidel matrix and—the original name—the (−1,1,0)-adjacency matrix) is the symmetric matrix with a row and column for each vertex, having 0 on the diagonal and, in the positions corresponding to vertices v_i and v_j, −1 if the vertices are adjacent and +1 if they are not. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by van Lint and Seidel (1966) and extensively exploited by Seidel and coauthors. It is the adjacency matrix of the signed complete graph 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.

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

See also

• Adjacency matrix

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. "Strongly Regular Graphs with (-1,1,0) Adjacency Matrix Having Eigenvalue 3." Lin. Alg. Appl. 1, 281-298, 1968.