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Non-trivial squares and Sidorenko's conjecture
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Non-trivial squares and Sidorenko's conjecture
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Paper
Non-trivial squares and Sidorenko's conjecture
2022
Overview
Let \\(t(H;G)\\) be the homomorphism density of a graph \\(H\\) into a graph \\(G\\). Sidorenko's conjecture states that for any bipartite graph \\(H\\), \\(t(H;G)\\geq t(K_2;G)^{|E(H)|}\\) for all graphs \\(G\\). It is already known that such inequalities cannot be certified through the sums of squares method when \\(H\\) is a so-called trivial square. In this paper, we investigate recent results about Sidorenko's conjecture and classify those involving trivial versus non-trivial squares. We then present some computational results. In particular, we categorize the bipartite graphs \\(H\\) on at most 7 edges for which \\(t(H;G)\\geq t(K_2;G)^{|E(H)|}\\) has a sum of squares certificate. We then discuss other limitations for sums of squares proofs beyond trivial squares.
Publisher
Cornell University Library, arXiv.org
Subject
