## Abstract

For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in L^{p}, 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.

Original language | English |
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Pages (from-to) | 919-929 |

Number of pages | 11 |

Journal | Discrete and Computational Geometry |

Volume | 67 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2022 Apr |

### Bibliographical note

Funding Information:Joonkyung Lee: Supported by ERC Consolidator Grant PEPCo 724903.

Funding Information:

Frederik Garbe, Jan Hladký: Supported by GAČR Project 18-01472Y. With institutional support RVO: 67985840.

Publisher Copyright:

© 2021, The Author(s).

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics