mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, the Seidel adjacency matrix of a simple undirected graph ''G'' is a

symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...

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 In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...

of ''G'' from the adjacency matrix of the

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...

of ''G''. The

multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by

J. H. van Lint Jacobus Hendricus ("Jack") van Lint (1 September 1932 – 28 September 2004) was a Dutch mathematician, professor at the Eindhoven University of Technology, of which he was rector magnificus from 1991 till 1996. He gained his Ph.D. from Utrecht U ...

and 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 In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex set ''X'', such that every (unordered) quadruple from ''X'' contains an even number of triples of the two-graph. A regular two-graph has the property that every ...

associated with ''G'' and ''K_G''. The eigenvalue properties of the Seidel matrix are valuable in the study of

strongly regular graph In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have comm ...

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. Algebraic graph theory Matrices {{combin-stub