Book contents
- Frontmatter
- Contents
- Preface
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Partial geometries
- 5 Strongly regular graphs with no triangles
- 6 Polarities of designs
- 7 Extensions of graphs
- 8 1-factorisations of K6
- 9 Codes
- 10 Cyclic codes
- 11 Threshold decoding
- 12 Finite geometries and codes
- 13 Self-orthogonal codes, designs and projective planes
- 14 Quadratic residue codes
- 15 Symmetry codes over GF(3)
- 16 Nearly perfect binary codes and uniformly packed codes
- 17 Association schemes
- References
- Index
2 - Strongly regular graphs
Published online by Cambridge University Press: 16 March 2010
- Frontmatter
- Contents
- Preface
- 1 A brief introduction to design theory
- 2 Strongly regular graphs
- 3 Quasi-symmetric designs
- 4 Partial geometries
- 5 Strongly regular graphs with no triangles
- 6 Polarities of designs
- 7 Extensions of graphs
- 8 1-factorisations of K6
- 9 Codes
- 10 Cyclic codes
- 11 Threshold decoding
- 12 Finite geometries and codes
- 13 Self-orthogonal codes, designs and projective planes
- 14 Quadratic residue codes
- 15 Symmetry codes over GF(3)
- 16 Nearly perfect binary codes and uniformly packed codes
- 17 Association schemes
- References
- Index
Summary
The theory of designs concerns itself with questions about subsets of a set (or relations between two sets) possessing a high degree of symmetry. By contrast, the large and amorphous area called ‘graph theory’ is mainly concerned with questions about ‘general’ relations on a set. This generality means that usually either the questions answered are too particular, or the results obtained are not powerful enough, to have useful consequences for design theory. There are some places where the two theories have interacted fruitfully; in the next five chapters, several of these areas will be considered. The unifying theme is provided by a class of graphs, the ‘strongly regular graphs’, introduced by Bose [16], whose definition reflects the symmetry inherent in t-designs. First, however, we shall mention an example of the kind of situation we shall not be discussing.
A graph consists of a finite set of vertices together with a set of edges, where each edge is a subset of the vertex set of cardinality 2. (In classical terms, our graphs are undirected, without loops or multiple edges.) As with designs, there is an alternative definition: a graph consists of a finite set of vertices and a set of edges, with an ‘incidence’ relation between vertices and edges, with the properties that any edge is incident with two vertices and any two vertices with at most one edge. Still another definition: a graph consists of a finite set of vertices together with a symmetric irreflexive binary relation (called adjacency) on the vertex set.
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- Graphs, Codes and Designs , pp. 16 - 24Publisher: Cambridge University PressPrint publication year: 1980
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