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Graph norms and Sidorenko’s conjecture

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Abstract

Let H and G be two finite graphs. Define h H (G) to be the number of homomorphisms from H to G. The function h H (·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for A G , the adjacency matrix of a graph G, we have h H (A G ) = h H (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h H (·)1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function h H (·)1/m is a norm.

We prove that h H (·)1/m is a norm if and only if a Hölder type inequality holds for H. We use this inequality to prove both positive and negative results, showing that h H (·)1/m is a norm for certain classes of graphs, and giving some necessary conditions on the structure of H when h H (·)1/m is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact, for such graphs we can prove statements that are much stronger than the assertion of Sidorenko’s conjecture.

We also investigate the h H (·)1/m norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the 2n-th Schatten-von Neumann norms.

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References

  1. G. R. Blakley and R. Prabir, A Hölder type inequality for symmetric matrices with nonnegative entries, Proceedings of the American Mathematical Society 16 (1965), 1244–1245.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Borgs, J. Chayes, L. Lovász, V. T. Sós and K. Vesztergombi, Counting graph homomorphisms, in Topics in Discrete Mathematics, Algorithms Combinatorics, Vol. 26, Springer, Berlin, 2006, pp. 315–371.

    Chapter  Google Scholar 

  3. F. Chung, R. Graham and R. M. Wilson, Quasi-random graphs, Combinatorica 9 (1989), 345–362.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Daudel, R. Lefebvre and C. Moser, Quantum Chemistry: Methods and Applications, Interscience Publishers, New York-London, 1959.

    MATH  Google Scholar 

  5. J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press, Cambridge, 1995.

    MATH  Google Scholar 

  6. P. Erdős and M. Simonovits, Compactness results in extremal graph theory, Combinatorica 2 (1982), 275–288.

    Article  MathSciNet  Google Scholar 

  7. T. Figiel, On the moduli of convexity and smoothness, Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica 56 (1976), 121–155.

    MATH  MathSciNet  Google Scholar 

  8. T. Figiel and G. Pisier, Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses, Comptes Rendus Mathématique. Académie des Sciences, Paris 279 (1974), 611–614.

    Google Scholar 

  9. P. Frankl and V. Rödl, Extremal problems on set systems, Random Structures Algorithms 20 (2002), 131–164.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Freedman, L. Lovász and A. Schrijver, Reflection positivity, rank connectivity, and homomorphism of graphs, Journal of the American Mathematical Society 20 (2007), 37–51 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, Journal d’Analyse Mathématique 34 (1979), 275–291.

    Article  MathSciNet  Google Scholar 

  12. T. Gowers, Hypergraph regularity and the multidimensional Szemerdi theorem, Annals of Mathematics. Second Series 166 (2007), 897–946.

    MATH  MathSciNet  Google Scholar 

  13. T. Gowers, A new proof of Szemerédi’s theorem for arithmetic progressions of length four, Geometric and Functional Analysis 8 (1998), 529–551.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Gowers, A new proof of Szemerédi’s theorem, Geometric and Functional Analysis 11 (2001), 465–588.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. Green and T. Tao, An inverse theorem for the Gowers U 3(G) norm, Proceedings of the Edinburgh Mathematical Society. Series II 51 (2008), 73–153.

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Green and T. Tao, New bounds for Szemeredi’s theorem, I: Progressions of length 4 in finite field geometries, Proceedings of the London Mathematical Society. Third Series 98 (2009), 365–392.

    Article  MATH  MathSciNet  Google Scholar 

  17. B. Green and T. Tao, New bounds for Szemeredi’s theorem, II: Progressions of length 4 in finite field geometries, in Analytic number theory, Cambridge University Press, Cambridge, 2009, pp. 180–204.

    Google Scholar 

  18. B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics. Second Series 167 (2008), 481–547.

    MATH  MathSciNet  Google Scholar 

  19. B. Host and B. Kra, Parallelepipeds, nilpotent groups, and Gowers norms, Bulletin de la Société Mathématique de France 136 (2008), 405–437.

    MATH  MathSciNet  Google Scholar 

  20. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Volume 97 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Springer-Verlag, Berlin, (1979). Function Spaces.

    MATH  Google Scholar 

  21. L. Lovász and B. Szegedy, Limits of dense graph sequences, Journal of Combinatorial Theory. Series B 96 (2006), 933–957.

    Article  MATH  MathSciNet  Google Scholar 

  22. B. Nagle, V. Rödl and M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms 28 (2006), 113–179.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Razborov, On the minimal density of triangles in graphs, Combinatorics, Probability and Computing 17 (2008), 603–618.

    MATH  MathSciNet  Google Scholar 

  24. V. Rödl, Some developments in Ramsey theory, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Journal of Mathematical Society of Japan, Tokyo 1991, pp. 1455–1466.

    Google Scholar 

  25. V. Rödl and J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms 25 (2004), 1–42.

    Article  MATH  MathSciNet  Google Scholar 

  26. V. Rödl and J. Skokan, Applications of the regularity lemma for uniform hypergraphs, Random Structures Algorithms 28 (2006), 180–194.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Schatten, The cross-space of linear transformations, Annals of Mathematics. Second Series 47 (1946), 73–84.

    MathSciNet  Google Scholar 

  28. R. Schatten and J. von Neumann, The cross-space of linear transformations. II, Annals of Mathematics. Second Series 47 (1946), 608–630.

    MathSciNet  Google Scholar 

  29. R. Schatten and J. von Neumann, The cross-space of linear transformations. III, Annals of Mathematics. Second Series 49 (1948), 557–582.

    MathSciNet  Google Scholar 

  30. A. Sidorenko, Inequalities for functionals generated by bipartite graphs, Diskretnaya Matematika 3 (1991), 50–65.

    MATH  Google Scholar 

  31. A. Sidorenko, A correlation inequality for bipartite graphs, Graphs and Combinatorics 9 (1993), 201–204.

    Article  MATH  MathSciNet  Google Scholar 

  32. G. Stell, Generating functionals and graphs, in Graph Theory and Theoretical Physics, Academic Press, London, 1967, pp. 281–300.

    Google Scholar 

  33. E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Mathematica Academiae Scientiarum Hungaricae 20 (1969), 89–104.

    Article  MATH  MathSciNet  Google Scholar 

  34. E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica 27 (1975), 199–245. Collection of articles in memory of Juriĭ Vladimirovič Linnik.

    MATH  MathSciNet  Google Scholar 

  35. T. Tao, The ergodic and combinatorial approaches to Szemerédi’s theorem, in Additive Combinatorics and Number Theory, CRM Proceedings Lecture Notes, Vol. 43, American Mathematical Society, Providence, RI, 2007, pp. 145–193.

    Google Scholar 

  36. N. Tomczak-Jaegermann, The moduli of smoothness and convexity and the Rademacher averages of trace classes S p(1 ≤ p < ∞), Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica 50 (1974), 163–182.

    MATH  MathSciNet  Google Scholar 

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  1. Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, Ontario, M5S 3G4, Canada

    Hamed Hatami

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  1. Hamed Hatami
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Hatami, H. Graph norms and Sidorenko’s conjecture. Isr. J. Math. 175, 125–150 (2010). https://doi.org/10.1007/s11856-010-0005-1

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