Abstract:Sidorenko's conjecture asserts that every bipartite graph has at least the expected homomorphism density in every graph of a given edge density. Motivated by Cayley-type formulations of Sidorenko-type inequalities, we study a two-sided correlation construction on finite groups.
Let $\Gamma$ be a finite group and let $f:\Gamma\to\mathbb{R}$ be a real-valued function. We define a directed kernel on $\Gamma$ by $$\mathcal C_f(x,y)=|\Gamma|^{-1}\sum_{a_1,a_2\in\Gamma:\, xa_1=a_2y} f(a_1)f(a_2)=\mathbb{E}_{z\in\Gamma} f(x^{-1}z)f(zy^{-1}).$$ When $f=\mathbf{1}_A$, this is the normalized size of the intersection $xA\cap Ay$.
We prove that, for every finite directed graph $F$, $$t(F,\mathcal C_f)\geq t(\overrightarrow{K_2},\mathcal C_f)^{e(F)}=(\mathbb{E}_{g\in\Gamma}f(g))^{2e(F)}.$$ Equivalently, if $W_f^\times(x,y)=f(xy)$ is the directed product Cayley kernel on $\Gamma$, then the directed $1$-subdivision of every finite directed graph satisfies the same homomorphism-density lower bound in $W_f^\times$.
From: Yuqi Zhao [view email]
[v1]
Mon, 22 Jun 2026 08:30:05 UTC (10 KB)
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