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