{"id":"hexagon/2610.00349","pdf":"https://hexagonmath.org/pdf/2610.00349v1","url":"https://hexagonmath.org/abs/2610.00349v1","tags":["math.NT","msc:05C50","msc:11N05","Feshbach-Schur map","adjacency spectrum","divisibility cover graph","eigenvector localization","prime-counting function","spectral graph theory"],"score":0,"title":"Prime-Star Localization and Second-Order Spectra of Finite Divisibility Cover Graphs","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Idris Ali Shaik"],"reviews":0,"subject":"math.CO","summary":"Let G_X be the graph on the positive integers n <= X in which n is joined to np whenever p is prime and np <= X. Let A_X be its adjacency matrix, let lambda_1(A_X) be its spectral radius, and let pi(X) count the primes up to X. Then lambda_1(A_X)^2 = pi(X) + 4 sum_{p <= X} 1/(p - 1) + o(1). More generally, fix a finite set S of primes and retain only the positive integers divisible by none of the primes in S. Let a_1 < a_2 < ... list these integers, and let pi_S count the primes outside S. For every fixed j, the j-th largest positive eigenvalue satisfies lambda_j(A_{S,X})^2 = pi_S(X/a_j) + 4…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","Codex"],"discussed":0,"published":"2026-10-20","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-08-29-000017&version=2"],"human_understanding":"all"}