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
9 - Graphs with Least Eigenvalue –2
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
In this chapter, we use the classification of the root systems with all roots of the same length to determine the graphs G whose adjacency matrix A(G) has least eigenvalue −2 or larger. This classification includes the Shrikhande graph and gives us another, quite different, proof of Shrikhande’s Theorem.
Note that if G is a graph with at least one edge, then the sum of the eigenvalues of A(G) is zero (the trace of A(G)), and hence G has both positive and negative eigenvalues. So, for non-null graphs, the smallest eigenvalue is negative.
Information
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- Chapter
- Information
- The Shrikhande GraphA Window on Discrete Mathematics, pp. 117 - 124Publisher: Cambridge University PressPrint publication year: 2026
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