Independence ratio and random eigenvectors in transitive graphs

TL;DR

This paper explores bounds on the independence ratio of transitive graphs using spectral properties and random eigenvectors, providing new lower bounds and constructions for infinite graphs.

Contribution

It introduces a novel spectral method using random eigenvectors to establish lower bounds on independence ratios in transitive graphs, extending to infinite cases.

Findings

Lower bound on independence ratio for 3-regular transitive graphs

Construction of factor of i.i.d. independent sets with high probability

The set of distributions of factor of i.i.d. processes is not closed in weak topology

Abstract

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\lambda_{\min}$ 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 least \[q=\frac{1}{2}-\frac{3}{4\pi}\arccos\biggl(\frac{1-\lambda _{\min}}{4}\biggr).\] The 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.