Construct and verify Ramanujan graphs for spectral edge expansion under degree constraints.
npx skills add https://github.com/plurigrid/asi --skill ramanujan-expander
Please help me install this Agent Skill. Skill: ramanujan-expander Source: https://github.com/plurigrid/asi/tree/main/skills/ramanujan-expander Command: npx skills add https://github.com/plurigrid/asi --skill ramanujan-expander
Ramanujan graphs are optimal spectral expanders; this skill covers their theory, edge-growth rules, and spectral centrality.
Check a given d-regular graph for λ₂ ≤ 2√(d-1) and use LPS construction to build Ramanujan graphs.
High-intent search queries and answers about installing and using this skill.
A d-regular graph is a Ramanujan expander if its second-largest eigenvalue λ₂ satisfies λ₂ ≤ 2√(d-1), the Alon-Boppana bound. Use eigenvalue computation to check this spectral condition and confirm optimal edge expansion properties.
LPS construction is an explicit method for building d-regular Ramanujan graphs using algebraic number theory. It generates optimal spectral expanders that achieve the theoretical lower bound on the second eigenvalue for degree-constrained networks.
Apply edge-growth rules that preserve the spectral bound λ₂ ≤ 2√(d-1) during expansion. Use non-backtracking methods and spectral verification at each growth step to ensure the graph remains a Ramanujan expander.
Use spectral expanders when you need networks with minimal mixing time and maximal edge connectivity under degree constraints. Ramanujan graphs optimize these properties, making them ideal for distributed systems, peer-to-peer networks, and fault-tolerant topology design.
Yes. Non-backtracking methods provide rigorous spectral property auditing by computing eigenvalues that detect community structure and verify expansion bounds. This complements traditional eigenvalue analysis for comprehensive network centrality checks.
LPS construction requires algebraic number-theoretic setup that scales with degree. Computational complexity and memory grow with network size; verification of the Alon-Boppana bound becomes expensive for very large graphs. Trade-offs exist between explicit construction and approximation methods.
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