Garbe, F;
Hladky, J;
Lee, J;
(2021)
Two Remarks on Graph Norms.
Discrete & Computational Geometry
10.1007/s00454-021-00280-w.
(In press).
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Abstract
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.
| Type: | Article |
|---|---|
| Title: | Two Remarks on Graph Norms |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00454-021-00280-w |
| Publisher version: | http://dx.doi.org/10.1007/s00454-021-00280-w |
| Language: | English |
| Additional information: | Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
| Keywords: | Graph norms, Graph limits, Graphons |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10124244 |
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