Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 1 The Erdős–Ko–Rado Theorem
- 2 Bounds on cocliques
- 3 Association schemes
- 4 Distance-regular graphs
- 5 Strongly regular graphs
- 6 The Johnson scheme
- 7 Polytopes
- 8 The exact bound in the EKR Theorem 135
- 9 The Grassmann scheme
- 10 The Hamming scheme
- 11 Representation theory
- 12 Representation theory of the symmetric group
- 13 Orbitals
- 14 Permutations
- 15 Partitions
- 16 Open problems
- Glossary: Symbols
- Glossary: Operations and relations
- References
- Index
4 - Distance-regular graphs
Published online by Cambridge University Press: 05 December 2015
- Frontmatter
- Dedication
- Contents
- Preface
- 1 The Erdős–Ko–Rado Theorem
- 2 Bounds on cocliques
- 3 Association schemes
- 4 Distance-regular graphs
- 5 Strongly regular graphs
- 6 The Johnson scheme
- 7 Polytopes
- 8 The exact bound in the EKR Theorem 135
- 9 The Grassmann scheme
- 10 The Hamming scheme
- 11 Representation theory
- 12 Representation theory of the symmetric group
- 13 Orbitals
- 14 Permutations
- 15 Partitions
- 16 Open problems
- Glossary: Symbols
- Glossary: Operations and relations
- References
- Index
Summary
A critical family of association schemes arise from distance-regular graphs. A graph is distance regular if for any two vertices u and v at distance k in the graph, the number of vertices w that are at distance i from u and j from v depends only on k, and not on the vertices u and v.
Suppose X is a graph with diameter d. The ith distance graph Xi of X has the same vertex set as X, and two vertices are adjacent in Xi if and only if they are distance i in X. For any graph X, the adjacency matrices of the distance graphs of X are called the distance matrices. The distance matrices for any graph are linearly independent and sum to J − I. It has been left as an exercise to show that if X is a distance-regular graph, then the distance matrices form an association scheme. Such an association scheme is called metric; these are discussed in detail in Section 4.1.
Let X be a graph and S a subset of the vertices of X. Denote by Si the set of vertices of X at distance i from S. The distance partition of X relative to S is the partition δS = { S1, …, Sr } of V (X). We say that S is a completely regular subset of X if δS is an equitable partition. If X is a distance-regular graph, then for any x ∈ V (X) the set S = { x} is completely regular. Moreover, the quotient graphs of X with respect to each δ{ x} are isomorphic. For any distance partition δS = { S1, …, Sr } of X, vertices in Si can only be adjacent to vertices in Si−1, Si and Si+1.
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- Erdõs–Ko–Rado Theorems: Algebraic Approaches , pp. 70 - 86Publisher: Cambridge University PressPrint publication year: 2015