Sidorenko-Type Inequalities for Even Subdivisions over Finite Abelian Groups

TL;DR

This paper proves Sidorenko's conjecture for even subdivisions of graphs within abelian Cayley graphs, using Fourier analysis and linear system solutions over finite abelian groups.

Contribution

It extends Sidorenko's conjecture to even subdivisions over abelian groups, a significant generalization in graph homomorphism inequalities.

Findings

Sidorenko's conjecture holds for even subdivisions in abelian Cayley graphs.

Homomorphism counts are reduced to linear system averages over finite abelian groups.

Fourier analysis techniques are effectively applied to prove the inequality.

Abstract

Sidorenko's conjecture asserts that every bipartite graph $H$ has the property that, for any host graph $G$, the homomorphism density from $H$ to $G$ is asymptotically at least as large as in a quasirandom graph with the same edge density as $G$. While the conjecture remains still very open, Szegedy showed that it suffices to verify the inequality when the host graph is a Cayley graph over a finite group. In this paper, we prove that Sidorenko's conjecture holds for all even subdivisions of arbitrary graphs when the host graph is a Cayley graph over an abelian group. That is, if each edge of a graph is replaced by a path of even length (allowing different lengths for different edges), then the resulting graph satisfies the Sidorenko's inequality in any abelian Cayley host graph. Our approach reduces the homomorphism count to the evaluation of certain averages over solution sets of linear systems over finite abelian groups, and proceeds using Fourier-analytic techniques.