Independence ratio and random eigenvectors in transitive graphs

Abstract

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum λmin of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a 3-regular transitive graph is at leastThe same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least q − o(1).We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.

On the almost eigenvectors of random regular graphs

“…The next proposition was proved by Harangi and Virág in [27]. Proposition 14.2 For |λ| ≤ 2 √ d − 1 the unique Gaussian wave Ψ λ is a weak limit of factor of i.i.d processes (but not a factor of i.i.d.…”

Spectral measures of factor of i.i.d. processes on vertex-transitive graphs

“…process if λ is the supremum of the spectrum. As for the d-regular tree G = T d (with d ≥ 3), Theorem 3 of Csóka, Gerencsér, Harangi and Virág (2015) states that Gaussian wave function (X v ) exists for all λ ∈ [−d, d]. Theorem 4 of the same paper says that (X v ) is a weak limit of factor of i.i.d.…”

“…According to Theorem 4 of Harangi and Virág (2015), such a process exists for all λ in the spectrum of the adjacency operator of G, but it is not a factor of i.i.d. process if λ is the supremum of the spectrum.…”

Entropy inequalities for factors of IID

“…These probabilities are independent and consequently the probability that a fixed coloring c is "good" for a random lift is the product of (12) with e running through E(G). To get the expected number of good colorings for a random lift we need to multiply this product by (13), and Lemma 5.6 follows. Finally we prove the claim.…”