Weakly Norming Graphs are Edge-Transitive

Abstract

Let be the class of bounded measurable symmetric functions on [0, 1]2. For a function h ∈ ℋ and a graph G with vertex set [v1,⌦,vn} and edge set E(G), define

$${t_G}(h) = \int \cdots \int {\prod\limits_{{\rm{\{ }}{v_i}{\rm{,}}{v_j}{\rm{\} }} \in E(G)} {h({x_i},{x_j})\;d{x_1} \cdots d{x_n}} } .$$

Answering a question raised by Conlon and Lee, we prove that in order for tG(∣h∣)1/∣E(G)∣ to be a norm on , the graph G must be edge-transitive.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    D. Conlon and J. Lee: Finite reflection groups and graph norms, Adv. Math.315 (2017), 130–165.

    MathSciNet  Article  Google Scholar 

  2. [2]

    F. Garbe, J. Hladký and J. Lee: Two remarks on graph norms, arXiv:1909.10987, 2019.

  3. [3]

    H. Hatami: Graph norms and Sidorenko’s conjecture, Israel J. Math.175 (2010), 125–150.

    MathSciNet  Article  Google Scholar 

  4. [4]

    D. Král’, T. Martins, P. P. Pach and M. Wrochna: The step Sidorenko property and nonnorming edge-transitive graphs, J. Combin. Theory Ser. A162 (2019), 34–54.

    MathSciNet  Article  Google Scholar 

  5. [5]

    J. Lee and B. Schülke: Convex graphon parameters and graph norms, Israel J. Math. (to appear), arXiv:1910.08454, 2019.

  6. [6]

    L. Lovász: Large networks and graph limits, volume 60 of Colloquium Publications, Amer. Math. Soc., 2012.

Download references

Acknowledgments

I would like to thank Joonkyung Lee for helpful discussions, and the three anonymous referees for their careful reading of the article and valuable suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alexander Sidorenko.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sidorenko, A. Weakly Norming Graphs are Edge-Transitive. Combinatorica 40, 601–604 (2020). https://doi.org/10.1007/s00493-020-4468-3

Download citation

Mathematics Subject Classification (2010)

  • 05C22
  • 05C60
  • 15A60