Which Generalized Petersen Graphs Satisfy the Riemann Hypothesis? The case k ≤ 10

It is classical that the Ihara zeta function of a connected cubic graph satisfies the Riemann Hypothesis if and only if the graph is Ramanujan, i.e., every eigenvalue of its adjacency matrix except ±3 has absolute value at most 2√2. The spectrum of the generalized Petersen graph P(n,k) is known in closed form by work of Gera and Stănică. Dudek bounded the spectral gap independently of the jump parameter k, which shows that there are at most finitely many generalized Petersen graphs that are Ramanujan. However, it was unknown which specific graphs had this property. In this paper we provide a complete answer to this question for every k ≤ 10. Our method eliminates all nested radicals in the bounds on the eigenvalues simultaneously, bringing the Ramanujan condition down to the nonnegativity of a single integer polynomial on a cyclotomic grid. This yields an explicit bound on n for each k (optimal for k = 2, 3, 4), a parity law showing that the bound is cut in half when k and n are both odd, and a full classification: exactly 247 graphs P(n,k) with k ≤ 10 satisfy the Riemann Hypothesis. All cases are decided by exact computation in cyclotomic integers, with no floating-point approximation.