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 >= e(H), denoted by t(H, W). One may then define corresponding functionals parallel to W parallel to(H) := vertical bar t(H, W)vertical bar 1/e(H) and parallel to W parallel to(r(H)) := t( H, vertical bar W vertical bar)(1/e(H)), and say that H is (semi-)norming if parallel to center dot parallel to H is a (semi-) norm and that H is weakly norming if parallel to center dot parallel to 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 parallel to center dot parallel to H, we prove that parallel to center dot parallel to r( H) 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.