Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Part 1 Biography
- Part II Graph Basics
- Part III Properties of the Shrikhande Graph
- Part IV The Shrikhande Graph in Context
- 6 Latin Squares
- 7 The Shrikhande Graph on the Torus
- 8 Root Systems
- 9 Graphs with Least Eigenvalue –2
- 10 Miscellanea
- 11 Further Reading
- References
- Index
8 - Root Systems
from Part IV - The Shrikhande Graph in Context
Published online by Cambridge University Press: 05 June 2026
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Part 1 Biography
- Part II Graph Basics
- Part III Properties of the Shrikhande Graph
- Part IV The Shrikhande Graph in Context
- 6 Latin Squares
- 7 The Shrikhande Graph on the Torus
- 8 Root Systems
- 9 Graphs with Least Eigenvalue –2
- 10 Miscellanea
- 11 Further Reading
- References
- Index
Summary
Root systems are beautiful geometric objects in Euclidean space, which crop up in many parts of mathematics, including Lie algebras, singularity theory, mathematical physics and graph theory.
In this chapter, we will use graph theory to discuss the famous ADE classification of root systems in which all roots have the same length. This will then be used to determine the graphs whose adjacency matrix has least eigenvalue −2 or greater in Chapter 9, where we will see a connection between the Shrikhande graph and exceptional root systems.
We work in the Euclidean space V = Rd, with the standard inner product.
Given a non-zero vector u ∈ V, there is a unique hyperplane Hu through the origin which is perpendicular to u.We define the reflection ru in this hyperplane to be the linear map which fixes every vector in Hu and maps u to −u.
Information
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- Chapter
- Information
- The Shrikhande GraphA Window on Discrete Mathematics, pp. 107 - 116Publisher: Cambridge University PressPrint publication year: 2026
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