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On Bounding the Diameter of a Distance-Regular Graph

Abstract

In this note we investigate how to use an initial portion of the intersection array of a distance-regular graph to give an upper bound for the diameter of the graph. We prove three new diameter bounds. Our bounds are tight for the Hamming d-cube, doubled Odd graphs, the Heawood graph, Tutte’s 8-cage and 12-cage, the generalized dodecagons of order (1, 3) and (1, 4), the Biggs-Smith graph, the Pappus graph, the Wells graph, and the dodecahedron.

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References

  1. S. Bang: Geometric distance-regular graphs without 4-claws, Linear Algebra Appl. 438 (2013), 37–46.

    MathSciNet  Article  Google Scholar 

  2. S. Bang: Diameter bounds for geometric distance-regular graphs, Discrete Math. 341 (2018), 253–260.

    MathSciNet  Article  Google Scholar 

  3. S. Bang: Geometric antipodal distance-regular graphs with a given smallest eigenvalue, Graphs Combin. 35 (2019), 1387–1399.

    MathSciNet  Article  Google Scholar 

  4. S. Bang, A. Dubickas, J. H. Koolen and V. Moulton: There are only finitely many distance-regular graphs of fixed valency greater than two, Adv. Math. 269 (2015), 1–55.

    MathSciNet  Article  Google Scholar 

  5. S. Bang, A. L. Gavrilyuk and J. H. Koolen: Distance-regular graphs without 4-claws, European J. Combin. 80 (2019), 120–142.

    MathSciNet  Article  Google Scholar 

  6. S. Bang, A. Hiraki and J. H. Koolen: Improving diameter bounds for distance-regular graphs, European J. Combin. 27 (2006), 79–89.

    MathSciNet  Article  Google Scholar 

  7. S. Bang, A. Hiraki and J. H. Koolen: Delsarte set graphs with small c2, Graphs Combin. 26 (2010), 147–162.

    MathSciNet  Article  Google Scholar 

  8. S. Bang, J. H. Koolen and V. Moulton: A bound for the number of columns l(c,a,b) in the intersection array of a distance-regular graph, European J. Combin. 24 (2003), 785–795.

    MathSciNet  Article  Google Scholar 

  9. N. Biggs: Algebraic graph theory, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2nd edition, 1993.

    MATH  Google Scholar 

  10. N. L. Biggs, A. G. Boshier and J. Shawe-Taylor: Cubic distance-regular graphs, J. London Math. Soc. 33 (1986), 385–394.

    MathSciNet  Article  Google Scholar 

  11. A. Blokhuis and A. E. Brouwer: Determination of the distance-regular graphs without 3-claws, Discrete Math. 163 (1997), 225–227.

    MathSciNet  Article  Google Scholar 

  12. A. E. Brouwer and J. H. Koolen: The distance-regular graphs of valency four, J. Algebraic Combin. 10 (1999), 5–24.

    MathSciNet  Article  Google Scholar 

  13. A. E. Brouwer, A. M. Cohen and A. Neumaier: Distance-regular graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1989.

    Google Scholar 

  14. A. E. Brouwer and A. Neumaier: A remark on partial linear spaces of girth 5 with an application to strongly regular graphs, Combinatorica 8 (1988), 57–61.

    MathSciNet  Article  Google Scholar 

  15. J. S. Caughman: IV, Intersection numbers of bipartite distance-regular graphs, Discrete Math. 163 (1997), 235–241.

    MathSciNet  Article  Google Scholar 

  16. B. V. C. Collins: The girth of a thin distance-regular graph, Graphs Combin. 13 (1997), 21–30.

    MathSciNet  Article  Google Scholar 

  17. E. van Dam, J. H. Koolen and H. Tanaka: Distance-regular graphs, Dynamic Surveys, Electron. J. Combin., 2016, http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS22/pdf.

  18. E. R. van Dam and W. H. Haemers: A characterization of distance-regular graphs with diameter three, J. Algebraic Combin. 6 (1997), 299–303.

    MathSciNet  Article  Google Scholar 

  19. R. M. Damerell and M. A. Georgiacodis: On the maximum diameter of a class of distance-regular graphs, Bull. London Math. Soc. 13 (1981), 316–322.

    MathSciNet  Article  Google Scholar 

  20. C. D. Godsil: Bounding the diameter of distance-regular graphs, Combinatorica 8 (1988), 333–343.

    MathSciNet  Article  Google Scholar 

  21. A. Hiraki: A characterization of the odd graphs and the doubled odd graphs with a few of their intersection numbers, European J. Combin. 28 (2007), 246–257.

    MathSciNet  Article  Google Scholar 

  22. Q. Iqbal, J. H. Koolen, J. Park and M. U. Rehman: Distance-regular graphs with diameter 3 and eigenvalue a2c3, Linear Algebra Appl. 587 (2020), 271–290.

    MathSciNet  Article  Google Scholar 

  23. A. A. Ivanov: Bounding the diameter of a distance-regular graph, Dokl. Akad. Nauk SSSR 271 (1983), 789–792.

    MathSciNet  Google Scholar 

  24. A. Jurišić, J. Koolen and Š. Miklavič: Triangle- and pentagon-free distance-regular graphs with an eigenvalue multiplicity equal to the valency, J. Combin. Theory Ser. B 94 (2005), 245–258.

    MathSciNet  Article  Google Scholar 

  25. A. Jurišić, J. Koolen and P. Terwilliger: Tight distance-regular graphs, J. Algebraic Combin. 12 (2000), 163–197.

    MathSciNet  Article  Google Scholar 

  26. J. H. Koolen: On subgraphs in distance-regular graphs, J. Algebraic Combin. 1 (1992), 353–362.

    MathSciNet  Article  Google Scholar 

  27. J. H. Koolen: The distance-regular graphs with intersection number a1 ≠ 0 and with an eigenvalue −1 − (b1/2), Combinatorica 18 (1998), 227–234.

    MathSciNet  Article  Google Scholar 

  28. J. H. Koolen and J. Park: Distance-regular graphs with a1 or c2 at least half the valency, J. Combin. Theory Ser. A 119 (2012), 546–555.

    MathSciNet  Article  Google Scholar 

  29. J. H. Koolen and J. Park: A relationship between the diameter and the intersection number c2 for a distance-regular graph, Des. Codes Cryptogr. 65 (2012), 55–63.

    MathSciNet  Article  Google Scholar 

  30. J. H. Koolen, J. Park and H. Yu: An inequality involving the second largest and smallest eigenvalue of a distance-regular graph, Linear Algebra Appl. 434 (2011), 2404–2412.

    MathSciNet  Article  Google Scholar 

  31. W. J. Martin: Scaffolds: a graph-based system for computations in Bose-Mesner algebras, Linear Algebra Appl. 619 (2021), 50–106.

    MathSciNet  Article  Google Scholar 

  32. B. Mohar and J. Shawe-Taylor: Distance-biregular graphs with 2-valent vertices and distance-regular line graphs, J. Combin. Theory Ser. B 38 (1985), 193–203.

    MathSciNet  Article  Google Scholar 

  33. A. Neumaier and S. Penjic: A unified view of inequalities for distance-regular graphs, part I, J. Combin. Theory Ser. B (2020), accepted for publication.

  34. A. Neumaier and S. Penjic: A unified view of inequalities for distance-regular graphs, part II, preprint, https://www.mat.univie.ac.at/~neum/papers.html.

  35. L. Pyber: A bound for the diameter of distance-regular graphs, Combinatorica 19 (1999), 549–553.

    MathSciNet  Article  Google Scholar 

  36. P. Safet: On the Terwilliger algebra of bipartite distance-regular graphs, University of Primorska, 2019, thesis (Ph.D.), http://osebje.famnit.upr.si/~penjic/research/.

  37. H. Suzuki: Bounding the diameter of a distance regular graph by a function of kd, Graphs Combin. 7 (1991), 363–375.

    MathSciNet  Article  Google Scholar 

  38. H. Suzuki: Bounding the diameter of a distance regular graph by a function of kd. II, J. Algebra 169 (1994), 713–750.

    MathSciNet  Article  Google Scholar 

  39. D. E. Taylor and R. Levingston: Distance-regular graphs, in: Combinatorial mathematics (Proc. Internat. Conf. Combinatorial Theory, Australian Nat. Univ., Canberra, 1977), Springer, Berlin, volume 686 of Lecture Notes in Math., 1978, 313–323.

    Chapter  Google Scholar 

  40. P. Terwilliger: The diameter of bipartite distance-regular graphs, J. Combin. Theory Ser. B 32 (1982), 182–188.

    MathSciNet  Article  Google Scholar 

  41. P. Terwilliger: Distance-regular graphs and (s,c,α,k)-graphs, J. Combin. Theory Ser. B 34 (1983), 151–164.

    MathSciNet  Article  Google Scholar 

  42. L. Y. Tsiovkina: Two new infinite families of arc-transitive antipodal distance-regular graphs of diameter three with λ = μ related to groups Sz(q) and 2G2(q), J. Algebraic Combin. 41 (2015), 1079–1087.

    MathSciNet  Article  Google Scholar 

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Acknowledgments

The authors thank the anonymous reviewers for helpful and constructive comments that contributed to improving the final version of the paper.

Safet Penjić acknowledges the financial support from the Slovenian Research Agency (research program P1-0285 and research project J1-1695).

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Neumaier, A., Penjić, S. On Bounding the Diameter of a Distance-Regular Graph. Combinatorica 42, 237–251 (2022). https://doi.org/10.1007/s00493-021-4619-1

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  • DOI: https://doi.org/10.1007/s00493-021-4619-1

Mathematics Subject Classification (2010)

  • 05E30
  • 05C12
  • 05C62