Non-trivial squares and Sidorenko's conjecture

TL;DR

This paper explores Sidorenko's conjecture, focusing on the role of trivial and non-trivial squares, providing classifications and computational results on sum of squares certificates for bipartite graphs with up to 7 edges.

Contribution

It classifies bipartite graphs with up to 7 edges regarding sum of squares certificates for Sidorenko's conjecture and discusses limitations of this proof method beyond trivial squares.

Findings

Identified bipartite graphs with up to 7 edges that admit sum of squares certificates.

Classified trivial versus non-trivial squares in the context of Sidorenko's conjecture.

Discussed limitations of sum of squares proofs beyond trivial squares.

Abstract

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.