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 pair of vertices lies in the same number of triples of the two-graph. Two-graphs have been studied because of their connection with equiangular lines and, for regular two-graphs, strongly regular graphs, and also finite groups because many regular two-graphs have interesting automorphism groups.
A two-graph is not a graph and should not be confused with other objects called 2-graphs in graph theory, such as 2-regular graphs.
Examples
On the set of vertices {1,...,6} the following collection of unordered triples is a two-graph:
- 123 124 135 146 156 236 245 256 345 346
This two-graph is a regular two-graph since each pair of distinct vertices appears together in exactly two triples.
Given a simple graph G = (V,E), the set of triples of the vertex set V whose induced subgraph has an odd number of edges forms a two-graph on the set V. Every two-graph can be represented in this way.[1]
As a more complex example, let T be a tree with edge set E. The set of all triples of E that are not contained in a path of T form a two-graph on the set E.[2]
Switching and graphs
A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed complete graphs.
Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and a nonadjacent pair becomes adjacent. The edges whose endpoints are both in the set, or both not in the set, are not changed. Graphs are switching equivalent if one can be obtained from the other by switching. An equivalence class of graphs under switching is called a switching class. Switching was introduced by van Lint & Seidel (1966) and developed by Seidel; it has been called graph switching or Seidel switching, partly to distinguish it from switching of signed graphs.
In the standard construction of a two-graph from a simple graph given above, two graphs will yield the same two-graph if and only if they are equivalent under switching.
To a graph G there corresponds a signed complete graph Σ on the same vertex set, whose edges are signed negative if in G and positive if not in G. Conversely, G is the subgraph of Σ that consists of all vertices and all negative edges. The two-graph of G can also be defined as the set of triples of vertices that support a negative triangle (a triangle with an odd number of negative edges) in Σ. Two signed complete graphs yield the same two-graph if and only if they are equivalent under switching.
Switching of G and of Σ are related: switching the same vertices in both yields a graph H and its corresponding signed complete graph.
If graphs G and H are in a same switching class, the multiset of eigenvalues of two Seidel adjacency matrices of G and H coincide.
Adjacency matrix
The adjacency matrix of a two-graph is the adjacency matrix of any corresponding signed complete graph; thus it is symmetric, is zero on the diagonal, and has entries ±1 off the diagonal. If G is the graph corresponding to Σ, this matrix is called the (0, −1, 1)-adjacency matrix or Seidel adjacency matrix of G. The Seidel matrix is a fundamental tool in the study of two-graphs, especially regular two-graphs.
Notes
- ^ Colburn & Dinitz 2007, p. 876, Remark 13.2
- ^ Cameron, P.J. (1994), "Two-graphs and trees", Discrete Mathematics, 127: 63–74 cited in Colburn & Dinitz 2007, p. 876, Construction 13.12
References
- Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989), Distance-Regular Graphs. Springer-Verlag, Berlin. Sections 1.5, 3.8, 7.6C.
- Colbourn, Charles J.; Dinitz, Jeffrey H. (2007), Handbook of Combinatorial Designs (2nd ed.), Boca Raton: Chapman & Hall/ CRC, pp. 875–882, ISBN 1-58488-506-8
- Chris Godsil and Gordon Royle (2001), Algebraic Graph Theory. Graduate Texts in Mathematics, Vol. 207. Springer-Verlag, New York. Chapter 11.
- 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.
- Taylor, D. E. (1977), Regular 2-graphs. Proceedings of the London Mathematical Society (3), vol. 35, pp. 257–274.
- 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.