Two remarks on graph norms

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00555479" target="_blank" >RIV/67985840:_____/22:00555479 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1007/s00454-021-00280-w" target="_blank" >https://doi.org/10.1007/s00454-021-00280-w</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00454-021-00280-w" target="_blank" >10.1007/s00454-021-00280-w</a>

Result language
angličtina
Original language name
Two remarks on graph norms
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GJ18-01472Y" target="_blank" >GJ18-01472Y: Graph limits and inhomogeneous random graphs</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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