Two Remarks on Graph Norms

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in $$ , $$ , denoted by t(H, W). One may then define corresponding functionals $$ and $$ , and say that H is (semi-)norming if $$ is a (semi-)norm and that H is weakly norming if $$ 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 $$ , we prove that $$ 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.