Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs

Abstract

Let GX be the graph on the positive integers n ≤ X in which n is joined to np whenever p is prime and np ≤ X. Let AX be its adjacency matrix and let π(X) count the primes up to X. The paper proves λ1(AX)2 = π(X) + 4 ∑p ≤ X 1/(p−1) + o(1). More generally, after deleting a fixed finite set of prime directions, every fixed positive extreme eigenvalue has the corresponding finite allowed-prime sum and an explicit divisor correction determined by its star centre. Prime-Mertens asymptotics recover the equivalent 4 log log X plus constant form. The proof splits the prime-labelled edges at the square-root threshold. The large-prime edges form a disjoint star forest. The remaining adjacency has sharp norm scale X1/4/sqrt(log X) but zero first-order matrix element on each star eigenvector, so the leading displacement is second order. The resulting finite prime-sum remainder has error O(1/log X) for fixed modes. The paper also proves hub–shell eigenvector localization, centre–shell adjacency evolution, and uniform constant-layer asymptotics in a logarithmic window of star centres.