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Two Remarks on Graph Norms.

Frederik Garbe1, Jan Hladký1, Joonkyung Lee2

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Discrete & Computational Geometry
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PubMed
Summary
This summary is machine-generated.

This study investigates graph norms, proving that weakly norming graphs are not uniformly convex or smooth. It also establishes that a graph without isolated vertices is norming if and only if its components are norming graphs.

Keywords:
Graph limitsGraph normsGraphons

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Area of Science:

  • Graph Theory
  • Functional Analysis
  • Combinatorics

Background:

  • Homomorphism density extends graph properties to symmetric functions.
  • The concepts of (semi-)norming and weakly norming graphs are introduced via associated functionals.

Purpose of the Study:

  • To investigate the uniform convexity and smoothness of functionals related to weakly norming graphs.
  • To establish a factorization property for (weakly) norming graphs concerning their components.

Main Methods:

  • Analysis of functionals derived from homomorphism densities.
  • Investigation of graph properties, specifically focusing on norming and weakly norming conditions.
  • Utilizing factorization techniques to simplify the study of graph norms.

Main Results:

  • Demonstrated that weakly norming graphs are neither uniformly convex nor uniformly smooth, addressing a question by Hatami.
  • Proved that a graph without isolated vertices is (weakly) norming if and only if all its connected components are (weakly) norming.

Conclusions:

  • The findings contribute to the understanding of (weakly) norming graphs and their associated functionals.
  • The factorization result simplifies the study of graph norms by allowing focus on connected graphs.
  • A correction to a previous theorem by Hatami regarding graph norms is provided.