Two remarks on graph norms

TL;DR

This paper investigates properties of graph norms derived from homomorphism densities, proving that certain weakly norming graphs lack uniform convexity or smoothness and establishing a factorization principle for (weakly) norming graphs.

Contribution

It answers a question by Hatami on convexity and smoothness of graph norms and proves a key factorization result for (weakly) norming graphs, correcting previous errors.

Findings

$ orm{.}_{r(H)}$ is not uniformly convex or smooth for weakly norming $H$

Every (weakly) norming graph without isolated vertices is a disjoint union of isomorphic (weakly) norming components

The factorization reduces the study of graph norms to connected graphs.

Abstract

For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t(H,W)$. One may then define corresponding functionals $\|W\|_{H}:=|t(H,W)|^{1/e(H)}$ and $\|W\|_{r(H)}:=t(H,|W|)^{1/e(H)}$ and say that $H$ is (semi-)norming if $\|.\|_{H}$ is a (semi-)norm and that $H$ is weakly norming if $\|.\|_{r(H)}$ is a norm. We obtain two results that contribute to the theory of (weakly) norming graphs. Firstly, answering a question of Hatami, who estimated the modulus of convexity and smoothness of $\|.\|_{H}$, we prove that $\|.\|_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. Secondly, we prove that every graph $H$ without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of $H$ when studying graph norms. In particular, we correct an error in the original statement of the aforementioned theorem by Hatami.