Seidel adjacency matrix
Encyclopedia
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 vi and vj, −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 graph
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, 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 vi and vj, −1 if the vertices are adjacent and +1 if they are not. The multiset
Multiset
In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
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
Two-graph
In mathematics, a two-graph is a set of triples chosen from a finite vertex set X, such that every quadruple from X contains an even number of triples of the two-graph. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two-graph...
associated with G.
The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graph
Strongly regular graph
In graph theory, a discipline within mathematics, a strongly regular graph is defined as follows. Let G = be a regular graph with v vertices and degree k...
s.