Title: | Lower bounds for integral functionals generated by bipartite graphs (English) |

Author: | Kaskosz, Barbara |

Author: | Thoma, Lubos |

Language: | English |

Journal: | Czechoslovak Mathematical Journal |

ISSN: | 0011-4642 (print) |

ISSN: | 1572-9141 (online) |

Volume: | 69 |

Issue: | 2 |

Year: | 2019 |

Pages: | 571-592 |

Summary lang: | English |

. | |

Category: | math |

. | |

Summary: | We study lower estimates for integral fuctionals for which the structure of the integrand is defined by a graph, in particular, by a bipartite graph. Functionals of such kind appear in statistical mechanics and quantum chemistry in the context of Mayer's transformation and Mayer's cluster integrals. Integral functionals generated by graphs play an important role in the theory of graph limits. Specific kind of functionals generated by bipartite graphs are at the center of the famous and much studied Sidorenko's conjecture, where a certain lower bound is conjectured to hold for every bipartite graph. In the present paper we work with functionals more general and lower bounds significantly sharper than those in Sidorenko's conjecture. In his 1991 seminal paper, Sidorenko proved such sharper bounds for several classes of bipartite graphs. To obtain his result he used a certain way of ``gluing'' graphs. We prove his inequality for a new class of bipartite graphs by defining a different type of gluing. (English) |

Keyword: | integral inequality |

Keyword: | bipartite graph |

Keyword: | graph homomorphism |

Keyword: | Sidorenko's conjecture |

MSC: | 05C35 |

MSC: | 26D15 |

idZBL: | Zbl 07088805 |

idMR: | MR3959965 |

DOI: | 10.21136/CMJ.2019.0453-17 |

. | |

Date available: | 2019-05-24T09:03:21Z |

Last updated: | 2020-02-27 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/147745 |

. | |

Reference: | [1] Bernardi, O.: Solution to a combinatorial puzzle arising from Mayer's theory of cluster integrals.Sémin. Lothar. Comb. 59 (2007), Article No. B59e, 10 pages. Zbl 1193.05012, MR 2465401 |

Reference: | [2] Bogachev, V. I.: Measure Theory. Vol. I, II.Springer, Berlin (2007). Zbl 1120.28001, MR 2267655, 10.1007/978-3-540-34514-5 |

Reference: | [3] Conlon, D., Fox, J., Sudakov, B.: An approximate version of Sidorenko's conjecture.Geom. Funct. Anal. 20 (2010), 1354-1366. Zbl 1228.05285, MR 2738996, 10.1007/s00039-010-0097-0 |

Reference: | [4] Conlon, D., Kim, J. H., Lee, C., Lee, J.: Some advances on Sidorenko's conjecture.J. London Math. Soc. 98 (2018), 593-608. Zbl 07000630, MR 3893193, 10.1112/jlms.12142 |

Reference: | [5] Conlon, D., Lee, J.: Finite reflection groups and graph norms.Adv. Math. 315 (2017), 130-165. Zbl 1366.05107, MR 3667583, 10.1016/j.aim.2017.05.009 |

Reference: | [6] Hatami, H.: Graph norms and Sidorenko's conjecture.Isr. J. Math. 175 (2010), 125-150. Zbl 1227.05183, MR 2607540, 10.1007/s11856-010-0005-1 |

Reference: | [7] Kaouche, A., Labelle, G.: Mayer and Ree-Hoover weights, graph invariants and bipartite complete graphs.P.U.M.A., Pure Math. Appl. 24 (2013), 19-29. Zbl 1313.05019, MR 3197094 |

Reference: | [8] Kim, J. H., Lee, C., Lee, J.: Two approaches to Sidorenko's conjecture.Trans. Am. Math. Soc. 368 (2016), 5057-5074. Zbl 1331.05220, MR 3456171, 10.1090/tran/6487 |

Reference: | [9] Král', D., Martins, T. L., Pach, P. P., Wrochna, M.: The step Sidorenko property and non-norming edge-transitive graphs.Available at https://arxiv.org/abs/1802.05007. MR 3873870 |

Reference: | [10] Labelle, G., Leroux, P., Ducharme, M. G.: Graph weights arising from Mayer's theory of cluster integrals.Sémin. Lothar. Comb. 54 (2005), Article No. B54m, 40 pages. Zbl 1188.82007, MR 2341745 |

Reference: | [11] Li, J. L., Szegedy, B.: On the logarithmic calculus and Sidorenko's conjecture.Available at https://arxiv.org/abs/1107.1153. |

Reference: | [12] Lovász, L.: Large Networks and Graph Limits.Colloquium Publications 60. American Mathematical Society, Providence (2012). Zbl 1292.05001, MR 3012035, 10.1090/coll/060 |

Reference: | [13] Mayer, J. E., Mayer, M. Göppert: Statistical Mechanics.J. Wiley and Sons, New York (1940),\99999JFM99999 66.1175.01. MR 0674819 |

Reference: | [14] Royden, H. L.: Real Analysis.Macmillan Publishing, New York (1988). Zbl 0704.26006, MR 1013117 |

Reference: | [15] Sidorenko, A. F.: Inequalities for functionals generated by bipartite graphs.Discrete Math. Appl. 2 (1991), Article No. 489-504 English. Russian original translation from Diskretn. Mat. 3 1991 50-65. Zbl 0787.05052, MR 1138091, 10.1515/dma.1992.2.5.489 |

Reference: | [16] Sidorenko, A.: A correlation inequality for bipartite graphs.Graphs Comb. 9 (1993), 201-204. Zbl 0777.05096, MR 1225933, 10.1007/BF02988307 |

Reference: | [17] Szegedy, B.: An information theoretic approach to Sidorenko's conjecture.Available at https://arxiv.org/abs/1406.6738. |

. |

*Fulltext not available
(moving wall
24 months)
*