Stewart, Arthur: The Directed Ihara Zeta of The Four-Gon Quiver

Zenodo. 2026The directed Ihara (Bowen-Lanford) zeta function of the four-gon quiver Q, the four-vertex six-arrow strongly connected quiver with adjacency spectrum determined by det(I − uA) = 1 − u² − u³ − u⁴, is computed explicitly. The zeta function is ζ_Q(u) = 1/[(1 + u)(1 − u − u³)]. Its four poles occupy three distinct moduli {0.6823, 1, 1.2107}, which do not lie on a common circle. Three structural facts follow. Q is non-regular, so the Ihara-Ramanujan axis |u| = 1/√q does not apply. The characteristic polynomial λ⁴ − λ² − λ − 1 is not self-reciprocal, so ζ_Q admits no functional equation of the form ζ(u) = χ(u) ζ(c/u). The Vieta relation u₀ · |u_c|² = 1 ties the Perron-root reciprocal to the complex pair's modulus, a three-scale structure arising from det(A) = −1. The computation is self-contained, and no claims beyond the stated determinant, factorization, and root structure are made. Keywords: directed Ihara zeta, Bowen-Lanford zeta, quiver, directed graph, adjacency matrix, non-regular digraph, Perron root, functional equation ( direct link )