This is an announcement for the paper "Graph norms and Sidorenko's conjecture" by Hamed Hatami.

Abstract: Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of homomorphisms from $H$ to $G$. The function $h_H(\cdot)$ extends in a natural way to a function from the set of symmetric matrices to $\mathbb{R}$ such that for $A_G$, the adjacency matrix of a graph $G$, we have $h_H(A_G)=h_H(G)$. Let $m$ be the number of edges of $H$. It is easy to see that when $H$ is the cycle of length $2n$, then $h_H(\cdot)^{1/m}$ is the $2n$-th Schatten-von Neumann norm. We investigate a question of Lov'{a}sz that asks for a characterization of graphs $H$ for which the function $h_H(\cdot)^{1/m}$ is a norm. We prove that $h_H(\cdot)^{1/m}$ is a norm if and only if a H"{o}lder type inequality holds for $H$. We use this inequality to prove both positive and negative results, showing that $h_H(\cdot)^{1/m}$ is a norm for certain classes of graphs, and giving some necessary conditions on the structure of $H$ when $h_H(\cdot)^{1/m}$ is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture. We also investigate the $h_H(\cdot)^{1/m}$ norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the $2n$-th Schatten-von Neumann norms.

Archive classification: math.FA math.CO

Mathematics Subject Classification: 46E30; 05C35

Remarks: to appear in Israel Journal of Mathematics

The source file(s), arxiv/normFinal.bbl: 6949 bytes arxiv/normFinal.tex: 57888 bytes arxiv/normFinal.toc: 1082 bytes, is(are) stored in gzipped form as 0806.0047.tar.gz with size 20kb. The corresponding postcript file has gzipped size 125kb.

Submitted from: hamed@cs.toronto.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/0806.0047

or

http://arXiv.org/abs/0806.0047

or by email in unzipped form by transmitting an empty message with subject line

uget 0806.0047

or in gzipped form by using subject line

get 0806.0047

to: math@arXiv.org.