Open Access
Translator Disclaimer
September 2015 Independence ratio and random eigenvectors in transitive graphs
Viktor Harangi, Bálint Virág
Ann. Probab. 43(5): 2810-2840 (September 2015). DOI: 10.1214/14-AOP952

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

\[q=\frac{1}{2}-\frac{3}{4\pi}\arccos(\frac{1-\lambda_{\min}}{4}).\] 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.

Citation

Download Citation

Viktor Harangi. Bálint Virág. "Independence ratio and random eigenvectors in transitive graphs." Ann. Probab. 43 (5) 2810 - 2840, September 2015. https://doi.org/10.1214/14-AOP952

Information

Received: 1 August 2013; Revised: 1 June 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1323.05101
MathSciNet: MR3395475
Digital Object Identifier: 10.1214/14-AOP952

Subjects:
Primary: 05C50 , 05C69 , 60G15

Keywords: ‎adjacency matrix , Factor of i.i.d. , independence ratio , Independent set , invariant Gaussian process , minimum eigenvalue , Regular graph , transitive graph

Rights: Copyright © 2015 Institute of Mathematical Statistics

JOURNAL ARTICLE
31 PAGES


SHARE
Vol.43 • No. 5 • September 2015
Back to Top