Book contents
- Frontmatter
- Contents
- PREFACE
- Introduction
- Generalized Steiner systems of type 3-(v, {4,6}, 1)
- Some remarks on D.R. Hughes' construction of M12 and its associated designs
- On k-sets of class [0,1,2,n]2 in PG(r,q)
- Covering graphs and symmetric designs
- Arcs and blocking sets
- Flat embeddings of near 2n-gons
- Codes, caps and linear spaces
- Geometries originating from certain distance-regular graphs
- Transitive automorphism groups of finite quasifields
- On k-sets of type (m,n) in projective planes of square order
- On k-sets of type (m,n) in a Steiner system S(2, l, v)
- Some translation planes of order 81
- A new partial geometry constructed from the Hoffman-Singleton graph
- Locally cotriangular graphs
- Coding theory of designs
- On shears in fixed-point-free affine groups
- On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture
- Cubic surfaces whose points all lie on their 27 lines
- Existence results for translation nets
- Translation planes having PSL(2,w) or SL(3,w) as a collineation group
- Sequenceable groups: a survey
- Polar spaces embedded in a projective space
- On relations among the projective geometry codes
- Partition loops and affine geometries
- Regular cliques in graphs and special 1½ designs
- Bericht über Hecke Algebren und Coxeter Algebren eindlicher Geometrien
- On buildings and locally finite Tits geometries
- Moufang conditions for finite generalized quadrangles
- Embedding geometric lattices in a projective space
- Coverings of certain finite geometries
- On class-regular projective Hjelmslev planes
- On multiplicity-free permutation representations
- On a characterization of the Grassmann manifold representing the lines in a projective space
- Affine subplanes of projective planes
- Point stable designs
- Other talks
- Participants
Geometries originating from certain distance-regular graphs
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- PREFACE
- Introduction
- Generalized Steiner systems of type 3-(v, {4,6}, 1)
- Some remarks on D.R. Hughes' construction of M12 and its associated designs
- On k-sets of class [0,1,2,n]2 in PG(r,q)
- Covering graphs and symmetric designs
- Arcs and blocking sets
- Flat embeddings of near 2n-gons
- Codes, caps and linear spaces
- Geometries originating from certain distance-regular graphs
- Transitive automorphism groups of finite quasifields
- On k-sets of type (m,n) in projective planes of square order
- On k-sets of type (m,n) in a Steiner system S(2, l, v)
- Some translation planes of order 81
- A new partial geometry constructed from the Hoffman-Singleton graph
- Locally cotriangular graphs
- Coding theory of designs
- On shears in fixed-point-free affine groups
- On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture
- Cubic surfaces whose points all lie on their 27 lines
- Existence results for translation nets
- Translation planes having PSL(2,w) or SL(3,w) as a collineation group
- Sequenceable groups: a survey
- Polar spaces embedded in a projective space
- On relations among the projective geometry codes
- Partition loops and affine geometries
- Regular cliques in graphs and special 1½ designs
- Bericht über Hecke Algebren und Coxeter Algebren eindlicher Geometrien
- On buildings and locally finite Tits geometries
- Moufang conditions for finite generalized quadrangles
- Embedding geometric lattices in a projective space
- Coverings of certain finite geometries
- On class-regular projective Hjelmslev planes
- On multiplicity-free permutation representations
- On a characterization of the Grassmann manifold representing the lines in a projective space
- Affine subplanes of projective planes
- Point stable designs
- Other talks
- Participants
Summary
Distance-regular graphs having intersection number c2 = 1 are point graphs of linear incidence systems. This simple observation plays a crucial role in both the existence proof of a regular near octagon “associated with the Hall-Janko group” whose point graph is the unique automorphic graph with intersection array {10, 8, 8, 2; 1, 1, 4, 5} and the non-existence proof of a distance-regular graph with intersection array {12, 8, 6,…; 1, 1, 2,…}. These results imply a partial answer to a problem put forward by Biggs in 1976.
BASIC NOTIONS
Let Γ = (V,E) be a connected graph of diameter d and let Γi(α) for α є V denote the set of vertices at distance i from α. We recall from [2] that Γ is distance-regular if for any i (0≤i≤d) the numbers bi = |Γi+1(α) ∩ Γ1 (β)| and ci = |Γi-1(α) ∩ Γ1 (β)| do not depend on the choice of α, β such that β є Γi(α). Of course, bd = 0 and cl = 1. Write k = | Γ1(α)|. The array {k, b1, b2,…, bd-1; c1, c2,…, cd} is called the intersection array of Γ.
The graph Γ is called distance-transitive if its automorphism group Aut(Γ) is transitive on each of the classes {{α, β} ⊃ V | β ∈Γi(α)} (0≤i≤d), and Γ is called automorphic (cf. [3]) whenever it is distance-transitive, not a complete graph or a line graph, and has an automorphism group which is primitive on V.
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- Information
- Finite Geometries and DesignsProceedings of the Second Isle of Thorns Conference 1980, pp. 81 - 87Publisher: Cambridge University PressPrint publication year: 1981
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