INDEPENDENCE RATIO AND RANDOM EIGENVECTORS IN TRANSITIVE GRAPHS

• 成果类型: Article
• 署名作者: Harangi, Viktor; Virag, Balint
• 署名单位: University of Toronto
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP952
• 发表日期: 2015
• 页码: 2810-2840
• 关键词: regular graphs; number; sets

摘要

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 = 1/2 - 3/4 pi arccos (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.