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Lin-Lu-Yau curvature and diameter of amply regular graphs

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUST-2021-0232
  • Received Date: 09 November 2021
  • Rev Recd Date: 16 November 2021
  • Publish Date: 31 December 2021
    Abstract

  • Abstract

    By Hall’s marriage theorem, we study lower bounds of the Lin-Lu-Yau curvature of amply regular graphs with girth 3 or 4 under different parameter restrictions.As a consequence,we show that each conference graph has positive Lin-Lu-Yau curvature.Our approach also provides a geometric proof of a known diameter estimates of amply regular graphs in the case of girth 4 and some special cases of girth 3.

    Abstract

    By Hall’s marriage theorem, we study lower bounds of the Lin-Lu-Yau curvature of amply regular graphs with girth 3 or 4 under different parameter restrictions.As a consequence,we show that each conference graph has positive Lin-Lu-Yau curvature.Our approach also provides a geometric proof of a known diameter estimates of amply regular graphs in the case of girth 4 and some special cases of girth 3.
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    • References(16)
  • [1]
    Chung F R K, Yau S T. Logarithmic Harnack inequalities. Math. Res. Lett., 1996,3(6): 793-812.
    [2]
    Ollivier Y. Ricci curvature of Markov chains on metric spaces. J Funct Anal, 2009, 256(3): 810-864.
    [3]
    Ollivier Y. A survey of Ricci curvature for metric spaces and Markov chains. In: Probabilistic Approach to Geometry. Tokyo: Math. Soc. Japan, 2010: 343-381.
    [4]
    Bauer F, Jost J, Liu S. Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator. Math. Res. Lett., 2012, 19(6): 1185-1205.
    [5]
    Jost J, Liu S. Ollivier’s Ricci curvature, local clustering and curvature-dimansioin inequalities on graphs. Discrete Comput. Goem., 2014, 51(2): 300-322.
    [6]
    Bhattacharya B B, Mukherjee S. Exact and asymptotic results on coarse Ricci curvature of graphs. Discrete Math., 2015, 338(1): 23-42.
    [7]
    Lin Y, Lu L, Yau S T. Ricci curvature of graphs. Tohoku Math. J. , 2011, 63: 605-627.
    [8]
    Lin Y, Yau S T. Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett., 2010, 17(2): 343-356.
    [9]
    Bourne D, Cushing D, Liu S, F, et al. Ollivier-Ricci idleness functions of graphs. SIAM J. Discrete Math., 2018, 32(2): 1408-1424.
    [10]
    Cushing D, Kamtue S, Koolen J, et al. Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature. Adv. Math., 2020, 369: 107188.
    [11]
    Bonini V, Carroll C, Dinh U, et al. Condensed Ricci curvature of complete and strongly regular graphs. Involve, 2020, 13 (4): 559-576.
    [12]
    Münch F, Wojciechowski R. Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds. Adv. Math., 2019, 356: 106759.
    [13]
    Cushing D, Kamtue S, Kangaslampi R, et al. Curvatures, graph products and Ricci flatness. J. Graph Theory, 2021, 96(4): 522-553.
    [14]
    Iceland E, Samorodnitsky A. On coset leader graphs of structured linear codes. Discrete Comput. Geom., 2020, 63(6): 560-576.
    [15]
    Brouwer A E, Cohen A M, Neumaier A. Distance-Regular Graphs. Berlin: Springer, 1989.
    [16]
    Bondy J A, Murty U S R. Graph Theory. Berlin: Springer, 2008.
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    [1]
    Chung F R K, Yau S T. Logarithmic Harnack inequalities. Math. Res. Lett., 1996,3(6): 793-812.
    [2]
    Ollivier Y. Ricci curvature of Markov chains on metric spaces. J Funct Anal, 2009, 256(3): 810-864.
    [3]
    Ollivier Y. A survey of Ricci curvature for metric spaces and Markov chains. In: Probabilistic Approach to Geometry. Tokyo: Math. Soc. Japan, 2010: 343-381.
    [4]
    Bauer F, Jost J, Liu S. Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator. Math. Res. Lett., 2012, 19(6): 1185-1205.
    [5]
    Jost J, Liu S. Ollivier’s Ricci curvature, local clustering and curvature-dimansioin inequalities on graphs. Discrete Comput. Goem., 2014, 51(2): 300-322.
    [6]
    Bhattacharya B B, Mukherjee S. Exact and asymptotic results on coarse Ricci curvature of graphs. Discrete Math., 2015, 338(1): 23-42.
    [7]
    Lin Y, Lu L, Yau S T. Ricci curvature of graphs. Tohoku Math. J. , 2011, 63: 605-627.
    [8]
    Lin Y, Yau S T. Ricci curvature and eigenvalue estimate on locally finite graphs. Math. Res. Lett., 2010, 17(2): 343-356.
    [9]
    Bourne D, Cushing D, Liu S, F, et al. Ollivier-Ricci idleness functions of graphs. SIAM J. Discrete Math., 2018, 32(2): 1408-1424.
    [10]
    Cushing D, Kamtue S, Koolen J, et al. Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature. Adv. Math., 2020, 369: 107188.
    [11]
    Bonini V, Carroll C, Dinh U, et al. Condensed Ricci curvature of complete and strongly regular graphs. Involve, 2020, 13 (4): 559-576.
    [12]
    Münch F, Wojciechowski R. Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds. Adv. Math., 2019, 356: 106759.
    [13]
    Cushing D, Kamtue S, Kangaslampi R, et al. Curvatures, graph products and Ricci flatness. J. Graph Theory, 2021, 96(4): 522-553.
    [14]
    Iceland E, Samorodnitsky A. On coset leader graphs of structured linear codes. Discrete Comput. Geom., 2020, 63(6): 560-576.
    [15]
    Brouwer A E, Cohen A M, Neumaier A. Distance-Regular Graphs. Berlin: Springer, 1989.
    [16]
    Bondy J A, Murty U S R. Graph Theory. Berlin: Springer, 2008.
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