Graph spectrum: eigenvalues of adjacency matrix A(G). For n-vertex graph: n real eigenvalues λ_1≥...≥λ_n. λ_1 = spectral radius. Σλ_i²=2|E|. Σλ_i³=6·triangles. Spectrum encodes structural information but doesn't determine G uniquely.
How to Use
Select graph:
- λ_i: All eigenvalues
- ρ: Spectral radius
- Info: Encoded properties
Spectral Information
From spectrum: |E|=Σλ²/2, triangles=Σλ³/6, bipartite iff spectrum symmetric about 0. Walks of length k: tr(A^k). Hoffman bound: α(G) ≤ n·(-λ_n)/(λ_1-λ_n). Expander graphs have small second eigenvalue.
Cospectral Graphs
Non-isomorphic graphs can share spectrum (cospectral). Smallest example: two trees on 6 vertices. Most graphs are determined by their spectrum (conjecture). Spectrum + other invariants often distinguish graphs.
Step-by-Step Instructions
- 1Select graph.
- 2Compute A(G).
- 3Find eigenvalues.
- 4Analyze spectrum.
- 5Extract properties.