Two Remarks on Graph Norms

Abstract

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in $L^p$ , $p\ge e\left(H\right)$ , denoted by t(H, W). One may then define corresponding functionals ${\Vert W\Vert }_H:={\left|t(H,W)\right|}^{1/e\left(H\right)}$ and ${\Vert W\Vert }_{r\left(H\right)}:=t{(H,|W\left|\right)}^{1/e\left(H\right)}$ , and say that H is (semi-)norming if ${\Vert ·\Vert }_H$ is a (semi-)norm and that H is weakly norming if ${\Vert ·\Vert }_{r\left(H\right)}$ 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 ${\Vert ·\Vert }_H$ , we prove that ${\Vert ·\Vert }_{r\left(H\right)}$ is neither 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 a negligence in the original statement of the aforementioned theorem by Hatami.

Keywords: Graph limits; Graph norms; Graphons.