Joonkyung Lee, University of Hamburg

Abstract

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable sym-metric functions W in Lp, p ≥ e(H), denoted by tH(W). One may then define corresponding functionalskWkH := |tH(W)|1/e(H) and kWkr(H):= tH(|W|)1/e(H) and say that H is (semi-)norming if k.kH is a (semi-)norm and that H is weakly norming if k.kr(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 K5,5 C10 and C6K2, are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that k.kr(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 k.kH. 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 Hladky, and  ́ Bjarne Schulke.