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arXiv · 1208.4863

Eigenvalues and Linear Quasirandom Hypergraphs

John Lenz·Dhruv Mubayi
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Abstract

Let p(k) denote the partition function of k. For each k >= 2, we describe a list of p(k)-1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness of Kohayakawa-R\"odl-Skokan and Conlon-H\`{a}n-Person-Schacht and the spectral approach of Friedman-Wigderson. For each of the quasirandom properties that are described, we define a largest and second largest eigenvalue. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon-H\`{a}n-Person-Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung-Graham-Wilson for graphs.

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John Lenz, Dhruv Mubayi. 2012-08-23. Eigenvalues and Linear Quasirandom Hypergraphs. https://arxiv.org/abs/1208.4863

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