ihara-zeta

The Ihara zeta function encodes non-backtracking closed walks in graphs. Construct the non-backtracking matrix B, then apply the Bass-Hashimoto determinant formula to compute ζ_G(u)^{-1}, which reveals graph structure through spectral analysis.

Non-backtracking walks exclude immediate reversals, filtering out trivial cycles. They reveal deeper structural patterns in graphs and form the foundation for spectral clustering, Ramanujan property detection, and distinguishing graph topology that standard walks obscure.

Apply Möbius-based filtering to the non-backtracking matrix eigenvalues to isolate primitive cycles and eliminate redundant closed walks. This extraction supports hypothesis testing and clustering by exposing fundamental cycle structure.

Yes. Analyze eigenvalues of the non-backtracking matrix via the Bass-Hashimoto determinant relation. Ramanujan graphs exhibit specific spectral bounds that this analysis reveals, useful for detecting optimal expansion properties in graph clustering.

The Bass-Hashimoto determinant formula ζ_G(u)^{-1} encodes graph connectivity through non-backtracking walks. Its spectral decomposition identifies clusters and community structure, enabling data-driven partitioning without explicit cluster assumptions.

Use non-backtracking methods when standard spectral approaches conflate short cycles with long-range structure. They excel at detecting fine-grained clustering, testing Ramanujan properties, and distinguishing community boundaries that adjacency matrices may blur.