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 , its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions in , , denoted by . One may then define corresponding functionals and and say that is (semi-)norming if is a (semi-)norm and that is weakly norming if 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 and , are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that is not uniformly convex nor uniformly smooth, provided that is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of . We also prove that every graph 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 when studying graph norms. Based on joint work with Frederik Garbe, Jan Hladký, and Bjarne Schülke.