ramanujan-expander

npx skills add majiayu000/claude-skill-registry --skill ramanujan-expander-plurigrid-asi-2 --agent claude-code

Same command for any agent — swap --agent for codex, cursor, copilot.

# Ramanujan Expander Skill > *"The Alon-Boppana bound is unbreakable. You cannot create a d-regular graph with λ₂ < 2√(d-1), even theoretically."* ## Overview Ramanujan graphs are **optimal spectral expanders** - they achieve the theoretical limit on eigenvalue separation. This skill provides: 1. **Alon-Boppana bound verification** - Prove your graph is optimal 2. **Edge growth rules** - Add edges while preserving Ramanujan property 3. **Centrality validity predicates** - Spectral methods for node importance 4. **Mixing time bounds** - O(log n) mixing from spectral gap ## The Alon-Boppana Bound ### Theorem (Alon-Boppana) For any d-regular graph G on n vertices: ``` λ₂(G) ≥ 2√(d-1) - o(1) as n → ∞ ``` where λ₂ is the second-largest eigenvalue of the adjacency matrix. ### Ramanujan Property A d-regular graph G is **Ramanujan** if: ``` |λ| ≤ 2√(d-1) for all eigenvalues λ ≠ ±d ``` This is the **tightest possible** spectral gap. ### Example: 4-Regular Graphs ``` d = 4 2√(d-1) = 2√3 ≈ 3.464 Maximum spectral gap = d - 2√(d-1) = 4 - 3.464 = 0.536 Your observed gap: ~0.54 ✓ (theoretically optimal) ``` ## Edge Growth Rules ### Rule 1: Preserve Regularity ```julia function add_edge_preserving

just ramanujan-verify graph.json     # Check Ramanujan property
just ramanujan-grow graph.json       # Add edges preserving property
just ramanujan-centrality graph.json # Compute spectral centrality
just ramanujan-mixing graph.json     # Estimate mixing time
just ramanujan-lps 5 13              # Generate LPS(5,13) graph