Two remarks on graph norms

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Two remarks on graph norms

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Two remarks on graph norms

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ18-01472Y" target="_blank" >GJ18-01472Y: Limity grafů a nehomogenní náhodné grafy</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Discrete & Computational Geometry

ISSN

0179-5376

e-ISSN

1432-0444

Svazek periodika

67

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

11

Strana od-do

919-929

Kód UT WoS článku

000618566600002

EID výsledku v databázi Scopus

2-s2.0-85100914327