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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


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.


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Viktor Harangi. Bálint Virág. "Independence ratio and random eigenvectors in transitive graphs." Ann. Probab. 43 (5) 2810 - 2840, September 2015.


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

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


Vol.43 • No. 5 • September 2015
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