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 J. J. Seidel 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.