ihara-zeta

npx skills add majiayu000/claude-skill-registry --skill ihara-zeta --agent claude-code

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

# Ihara Zeta Function Skill > *"The Ihara zeta function encodes all non-backtracking closed walks - the 'prime cycles' of a graph."* ## Overview The Ihara zeta function generalizes the Riemann zeta function to graphs: 1. **Prime cycles** - Non-backtracking closed walks (graph analog of primes) 2. **Determinant formula** - ζ_G(u)^{-1} = det(I - uB) relation 3. **Ramanujan connection** - Riemann Hypothesis analog for graphs 4. **Non-backtracking matrix** - Central object for spectral clustering ## Definition ### Ihara Zeta Function For a graph G, the **Ihara zeta function** is: ``` ζ_G(u) = ∏_{[C]} (1 - u^{|C|})^{-1} ``` where: - Product is over equivalence classes [C] of **primitive** closed non-backtracking walks - |C| is the length of the cycle - Primitive = not a power of a shorter cycle ### Non-Backtracking Walk A walk `v₀ → v₁ → v₂ → ... → vₖ` is **non-backtracking** if: ``` vᵢ₊₁ ≠ vᵢ₋₁ for all i ``` (Never immediately return to the previous vertex) ## The Determinant Formula ### Bass-Hashimoto Formula ``` ζ_G(u)^{-1} = (1 - u²)^{|E| - |V|} · det(I - uB) ``` where **B** is the non-backtracking matrix. ### Non-Backtracking Matrix Indexed by **directed edges** (e, f) where head(e

just ihara-zeta graph.json           # Compute zeta function
just ihara-primes graph.json 10      # List prime cycles up to length 10
just ihara-grh graph.json            # Check Graph Riemann Hypothesis
just ihara-cluster graph.json 3      # Non-backtracking clustering
just ihara-spectrum graph.json       # Eigenvalues of B matrix