Book contents
- Frontmatter
- Contents
- Preface
- 1 Combinatorics of finite sets
- 2 Introduction to designs
- 3 Vector spaces over finite fields
- 4 Hadamard matrices
- 5 Resolvable designs
- 6 Symmetric designs and t-designs
- 7 Symmetric designs and regular graphs
- 8 Block intersection structure of designs
- 9 Difference sets
- 10 Balanced generalized weighing matrices
- 11 Decomposable symmetric designs
- 12 Subdesigns of symmetric designs
- 13 Non-embeddable quasi-residual designs
- 14 Ryser designs
- Appendix
- References
- Index
7 - Symmetric designs and regular graphs
Published online by Cambridge University Press: 26 February 2010
- Frontmatter
- Contents
- Preface
- 1 Combinatorics of finite sets
- 2 Introduction to designs
- 3 Vector spaces over finite fields
- 4 Hadamard matrices
- 5 Resolvable designs
- 6 Symmetric designs and t-designs
- 7 Symmetric designs and regular graphs
- 8 Block intersection structure of designs
- 9 Difference sets
- 10 Balanced generalized weighing matrices
- 11 Decomposable symmetric designs
- 12 Subdesigns of symmetric designs
- 13 Non-embeddable quasi-residual designs
- 14 Ryser designs
- Appendix
- References
- Index
Summary
Incidence relations defining designs and incidence relations induced by designs can sometimes be expressed in terms of graphs. Such graphs usually have a high degree of regularity reflecting the regularity of the corresponding designs.
Strongly regular graphs
Let N be an incidence matrix of a symmetric (v, k, λ)-design. If N is symmetric with zeros on the diagonal, it serves as an adjacency matrix of a graph Γ of order v. This graph is regular of degree k, and for any distinct vertices x and y of Γ, there are exactly λ vertices which are adjacent to both x and y.
If N is a symmetric incidence matrix of a symmetric (v, k, λ)-design with ones on the diagonal, then N – I serves as an adjacency matrix of a regular graph of order v and degree k – 1. For any distinct vertices x and y of this graph, the number of vertices that are adjacent to both x and y is equal to λ – 2 if x and y are adjacent and is equal to λ otherwise.
The graphs we have just described are special cases of strongly regular graphs.
- Type
- Chapter
- Information
- Combinatorics of Symmetric Designs , pp. 212 - 246Publisher: Cambridge University PressPrint publication year: 2006