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 $L^p$, $p\ge e(H)$, denoted by $t(H,W)$. One may then define corresponding functionals $\Vert W{\Vert }_H:=|t(H,W){|}^{1/e(H)}$ and $\Vert W{\Vert }_{r(H)}:=t(H,|W|{)}^{1/e(H)}$ and say that $H$ is (semi-)norming if $\Vert .{\Vert }_H$ is a (semi-)norm and that $H$ is weakly norming if $\Vert .{\Vert }_{r(H)}$ 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 $\Vert .{\Vert }_H$, we prove that $\Vert .{\Vert }_{r(H)}$ 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.

10 pages

References in corpus (2)

• Sidorenko's conjecture for blow-ups
• Cut distance identifying graphon parameters over weak* limits

Cited by in corpus (1)

• On graph norms for complex-valued functions