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 ${\lambda }_{\min }$ 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 \[q=\frac{1}{2}-\frac{3}{4π}\arccos\biggl(\frac{1-λ_{\min}}{4}\biggr).\] 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.

Published at http://dx.doi.org/10.1214/14-AOP952 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (6)

• Suboptimality of local algorithms for a class of max-cut problems
• Factor of iid percolation on trees
• Spectral measures of factor of i.i.d. processes on vertex-transitive graphs
• Entropy and expansion
• Correlation bound for distant parts of factor of IID processes
• The threshold for SDP-refutation of random regular NAE-3SAT