Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- I Basic results from designs
- II Strongly regular graphs and partial geometries
- III Basic results on quasi-symmetric designs
- IV Some configurations related to strongly regular graphs and quasi-symmetric designs
- V Strongly regular graphs with strongly regular decompositions
- VI The Witt designs
- VII Extensions of symmetric designs
- VIII Quasi-symmetric 2-designs
- IX Towards a classification of quasi-symmetric 3-designs
- X Codes and quasi-symmetric designs
- References
- Index
II - Strongly regular graphs and partial geometries
Published online by Cambridge University Press: 05 May 2010
- Frontmatter
- Contents
- Preface
- Acknowledgments
- I Basic results from designs
- II Strongly regular graphs and partial geometries
- III Basic results on quasi-symmetric designs
- IV Some configurations related to strongly regular graphs and quasi-symmetric designs
- V Strongly regular graphs with strongly regular decompositions
- VI The Witt designs
- VII Extensions of symmetric designs
- VIII Quasi-symmetric 2-designs
- IX Towards a classification of quasi-symmetric 3-designs
- X Codes and quasi-symmetric designs
- References
- Index
Summary
Our aim in this chapter is to gather together some basic results from strongly regular graphs and partial geometries. These topics have had a profound influence in the area of combinatorial designs after Bose's classical paper of 1963. The results of the first two chapters will provide the necessary background for later chapters. We refer to Harary for the necessary background in graph theory. Marcus and Mine will generally suffice for details of matrix results used. For further applications of matrix tools in a variety of problems on designs, we refer to M.S. Shrikhande.
Let Γ be a finite undirected graph on n vertices. The adjacency matrix A of Γ is a square matrix of size n. The diagonal entries of A are zero and for i ≠ j, the (i, j) entry of A is 1 or 0 according as the vertices i and j are joined by an edge or not. There are other types of adjacency matrices used. For example in Seidel, or Goethals and Seidel, a (0, ± 1) adjacency matrix is used.
A graph Γ is called regular of valency a if A has constant row sum a. The adjacency matrix reflects many other graphical properties of Γ.
We now state two basic definitions. Firstly, a matrix A is permutationally congruent to a matrix B if there is a permutation matrix P such that A = PtBP.
- Type
- Chapter
- Information
- Quasi-symmetric Designs , pp. 17 - 33Publisher: Cambridge University PressPrint publication year: 1991