IBS/KAIST Joint Discrete Math Seminar

On some properties of graph norms

Joonkyung Lee (이준경)

Department of Mathematics, University of Hamburg, Germany

Department of Mathematics, University of Hamburg, Germany

2019/10/22 Tue 4:30PM-5:30PM (Room B232, IBS)

For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t_H(W)$. One may then define corresponding functionals $\|W\|_{H}:=|t_H(W)|^{1/e(H)}$ and $\|W\|_{r(H)}:=t_H(|W|)^{1/e(H)}$ and say that $H$ is (semi-)norming if $\|.\|_{H}$ is a (semi-)norm and that $H$ is weakly norming if $\|.\|_{r(H)}$ 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 $C_6\square K_2$, are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that $\|.\|_{r(H)}$ 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 $\|.\|_{H}$. 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.