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We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector(More)
A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum λmin 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 =(More)
Consider a 1 ,. .. , a n ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of a j-periodic functions, i.e., f = f 1 +· · ·+f n with ∆ a j f (x) := f (x+a j)−f (x) = 0. We show that f has such a decomposition if and only if for all partitions B 1 ∪B 2 ∪· · ·∪B N = {a 1 ,. .. , a n } with B j consisting of(More)
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