Conjugacy Class Averages and Sidorenko’s Conjecture
ABSTRACT
Sidorenko's conjecture asserts that for every bipartite graph $H$ and every graph $G$,
\[
t(H,G) \geq t(K_2,G)^{e(H)}.
\]
A result of Szegedy shows that, in order to prove the conjecture, it suffices to verify the corresponding inequality on a special family of highly symmetric bipartite Cayley-type hosts arising from symmetric groups. Motivated by this reduction, we study Cayley-type bipartite kernels associated with functions on finite groups and their conjugacy class averages.
We prove that, for a fixed bipartite graph $H$, if the $H$-density of every nonnegative bipartite Cayley kernel is at least that of its conjugacy class average, then $H$ is Sidorenko. We also establish a Sidorenko-type inequality for $1$-subdivisions of arbitrary graphs in conjugacy-averaged Cayley kernels arising from arbitrary real-valued functions on finite groups.
