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
V - Strongly regular graphs with strongly regular decompositions
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
In the previous Chapter IV, we looked at the possibility of a strongly regular graph G containing vertex subsets which define a design or a PBIBD when the adjacency is suitably translated into incidence. A fairly general set-up was given in the notion of an SPBIBD of the last chapter. Essentially, the block graphs of interesting incidence structures are, of course, strongly regular but the combination of relations between points, blocks and point-blocks sometimes produces larger strongly regular graphs, and the procedure is often reversible. In a recent paper Haemers and Higman obtained an elegant combinatorial generalization of these ideas, where a graph is partitioned into two strongly regular graphs but these graphs are not necessarily assumed to arise out of a design or a PBIBD. The present chapter is almost entirely based on the paper of Haemers and Higman. Here we consider a strongly regular graph г0 whose vertex set V0 can be written as a disjoint union of two sets V1 and V2 such that the induced graph гi on Vi, i = 1, 2 has some nice properties. These nice properties include regularity, strong regularity, being a complete graph or its complement, etc.
Definition 5.1. Let m < n and let ρ1 ≥ ρ2 ≥ … ≥ ρn and σ1 ≥ σ2 ≥ … ≥ σm be two sequences of real numbers.
- Type
- Chapter
- Information
- Quasi-symmetric Designs , pp. 82 - 98Publisher: Cambridge University PressPrint publication year: 1991