Two remarks on graph norms

0555479 - MÚ 2023 RIV US eng J - Journal Article
Garbe, Frederik - Hladký, Jan - Lee, J.
Two remarks on graph norms.
Discrete & Computational Geometry. Roč. 67, č. 3 (2022), s. 919-929. ISSN 0179-5376. E-ISSN 1432-0444
R&D Projects: GA ČR(CZ) GJ18-01472Y
Institutional support: RVO:67985840
Keywords : graph limits * graph norms * graphons
OECD category: Pure mathematics
Impact factor: 0.8, year: 2022
Method of publishing: Open access
https://doi.org/10.1007/s00454-021-00280-w

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in Lp, p≥ 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 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 ‖·‖H, we prove that ‖·‖r(H) is neither 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 a negligence in the original statement of the aforementioned theorem by Hatami.
Permanent Link: http://hdl.handle.net/11104/0329991