Joonkyung Lee (이준경), On some properties of graph norms

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 some results that contribute to the theory of (weakly) norming graphs. Firstly, we show that ‘twisted’ blow-ups of cycles, which include $K_{5,5}\setminus C_{10}$ and $C_6\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.1.0/svg/25fb.svg">}{K}_2$, are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that $\Vert .{\Vert }_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of $\Vert .{\Vert }_H$. We also 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. Based on joint work with Frederik Garbe, Jan Hladký, and Bjarne Schülke.