# Hadwiger-Nelson Spectral Database

**Date:** 2026-07-04T08:50:51Z
**SHA-256:** `6af248bdf51d4a90b6e286e4decc33f60b153b4b162c677ecfbaf9f12b20641c`
**Includes de Grey:** False

## Spectral Bounds Comparison

| Graph | n | e | λ_max | λ_min | Hoffman | χ_Hoff | Welch-Wynn | χ_WW | Known χ | Gap |
|-------|---|---|-------|------|---------|--------|------------|------|---------|-----|
| Moser spindle | 7 | 12 | 3.6458 | -2.0000 | 2.8229 | 3 | N/A | None | 4 | gap=1 |
| Golomb graph (8 vertices, 10 edges) | 8 | 10 | 2.6813 | -2.3234 | 2.1541 | 3 | N/A | None | 4 | gap=1 |
| Empty graph (10v baseline) | 10 | 0 | 0.0000 | 0.0000 | inf | None | N/A | None | 1 | ? |
| Path graph P10 (baseline) | 10 | 9 | 1.9190 | -1.9190 | 2.0000 | 2 | N/A | None | 2 | tight |
| Cycle C5 (baseline) | 5 | 5 | 2.0000 | -1.6180 | 2.2361 | 3 | N/A | None | 3 | tight |
| Complete K4 (baseline) | 4 | 6 | 3.0000 | -1.0000 | 4.0000 | 4 | N/A | None | 4 | tight |

## Key Findings

1. **Hoffman bound** (χ ≥ 1 - λ_max/λ_min): classic spectral bound
2. **Welch-Wynn bound** (χ ≥ n/(n - λ_max)): another spectral bound
3. **Gap**: difference between best spectral bound and known chromatic number
   - 'tight' = spectral bound matches known χ
   - 'gap=N' = spectral bound is N below known χ

The gap measures how much chromatic information is NOT captured
by the spectrum. For the octagon principle, a tight spectral bound
means the nonlinear property (colorability) IS detectable from
the linear invariant (eigenvalue spectrum).
