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Linear Programming Bounds for Regular Graphs

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Abstract

Delsarte et al. (Geom Dedicata 6:363–388, 1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some “dual” technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected k-regular graph satisfying \(g>2d-1\) has the minimum second-largest eigenvalue of all k-regular graphs of the same size, where d is the number of distinct non-trivial eigenvalues, and g is the girth. The known graphs satisfying \(g>2d-1\) are Moore graphs, incidence graphs of regular generalized polygons of order (s, s), triangle-free strongly regular graphs, and the odd graph of degree 4.

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References

  1. Abiad, A., Van Dam, E.R., Fiol, M.A.: Some spectral and quasi-spectral characterizations of distance-regular graphs. arXiv:1404.3973

  2. Alon, N.: Eigenvalues and expanders. Combinatorica 6, 83–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Milman, V.D.: \(\lambda _1\), isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38, 73–88 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bachoc, C.: Linear programming bounds for codes in Grassmannian spaces. IEEE Trans. Inf. Theory 52, 2111–2125 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bachoc, C., Vallentin, F.: Optimality and uniqueness of the \((4,10,1/6)\) spherical code. J. Combin. Theory Ser. A 116(1), 195–204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bannai, E., Ito, T.: Algebraic Combinatorics I: Association Schemes. Benjamin/Cummings, Menlo Park (1984)

    MATH  Google Scholar 

  7. Bannai, E., Ito, T.: On finite Moore graphs. J. Fac. Sci. Univ. Tokyo Ser. A 20, 191–208 (1973)

    MathSciNet  MATH  Google Scholar 

  8. Barg, A., Purkayastha, P.: Bounds on ordered codes and orthogonal arrays. Mosc. Math. J. 9, 211–243 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Benson, C.T.: Minimal regular graphs of girths eight and twelve. Can. J. Math. 18, 1091–1094 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brouwer, A.E.: The uniqueness of the strongly regular graph on \(77\) points. J. Graph Theory 7(4), 455–461 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  12. Brouwer, A.E., Haemers, W.H.: The Gewirtz graph: an exercise in the theory of graph spectra. Eur. J. Combin. 14, 397–407 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cohen, A.M., Tits, J.: On generalized hexagons and a near octagon whose lines have three points. Eur. J. Combin. 6, 13–27 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cohn, H., Kumar, A.: Universally optimal distribution of points on spheres. J. Am. Math. Soc. 20(1), 99–184 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer, New York (1999)

    Book  MATH  Google Scholar 

  16. Damerell, R.M.: On Moore graphs. Camb. Philos. Soc. 74, 227–236 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 1–97 (1973)

    MathSciNet  Google Scholar 

  19. Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6(3), 363–388 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284, 787–794 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ericson, T., Zinoviev, V.: Codes on Euclidean Spheres. North-Holland Mathematical Library, 63, North-Holland Publishing Co., Amsterdam (2001)

    MATH  Google Scholar 

  22. Feit, W., Higman, G.: The nonexistence of certain generalized polygons. J. Algebra 1, 114–131 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gewirtz, A.: Graphs with maximal even girth. Can. J. Math. 21, 915–934 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  24. Godsil, C.D.: Problems in algebraic combinatorics. Electron. J. Combin. 2 #F1, 20 (1995)

  25. Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics, vol. 207. Springer, New York (2001)

    Book  Google Scholar 

  26. Hall Jr, M.: Uniqueness of the projective plane with \(57\) points. Proc. Am. Math. Soc. 4, 912–916 (1953)

    MATH  Google Scholar 

  27. Hall Jr, M., Swift, J.D., Killgrove, R.: On projective planes of order nine. Math. Tables Aids Comput. 13, 233–246 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hall Jr, M., Swift, J.D., Walker, R.J.: Uniqueness of the projective plane of order eight. Math. Tables Aids Comput. 10, 186–194 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  29. Higman, D.G., Sims, C.C.: A simple group of order 44,352,000. Math. Z. 105, 110–113 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hoffman, A.J.: On the polynomial of a graph. Am. Math. Mon. 70, 30–36 (1963)

    Article  MATH  Google Scholar 

  31. Hoffman, A.J., Singleton, R.R.: On Moore graphs with diameters 2 and 3. IBM J. Res. Dev. 4, 497–504 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43, 439–561 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hora, A., Obata, N.: Quantum Probability and Spectral Analysis of Graphs. Theoretical and Mathematical Physics. Springer, Berlin (2007)

    Google Scholar 

  34. Kabatiansky, G.A., Levenshtein, V.I.: Bounds for packings on a sphere and in space. Probl. Peredachi Inf. 14, 3–25 (1978)

    Google Scholar 

  35. Koledin, T., Stanić, Z.: Regular graphs whose second largest eigenvalue is at most 1. Novi Sad J. Math. 43, 145–153 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Levenshtein, V.I.: Designs as maximum codes in polynomial metric spaces. Acta Appl. Math. 29, 1–82 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. MacInnes, C.R.: Finite planes with less than eight points on a line. Am. Math. Mon. 14, 171–174 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  38. Martin, W.J., Tanaka, H.: Commutative association schemes. Eur. J. Combin. 30, 1497–1525 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Moon, A.: Characterization of the odd graphs \(O_k\) by parameters. Discrete Math. 42(1), 91–97 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  40. Musin, O.R.: The kissing number in four dimensions. Ann. Math. 168(1), 1–32 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Musin, O.R., Tarasov, A.S.: The strong thirteen spheres problem. Discrete Comput. Geom. 48(1), 128–141 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Odlyzko, A.M., Sloane, N.J.A.: New bounds on the number of unit spheres that can touch a unit sphere in \(n\) dimensions. J. Combin. Theory Ser. A 26(2), 210–214 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  43. Payne, S.E.: All generalized quadrangles of order 3 are known. J. Combin. Theory Ser. A 18, 203–206 (1975)

    Article  MATH  Google Scholar 

  44. Payne, S.E.: Generalized quadrangles of order 4. I. J. Combin. Theory Ser. A 22(3), 267–279 (1977)

    Article  MATH  Google Scholar 

  45. Payne, S.E.: Generalized quadrangles of order 4. II. J. Combin. Theory Ser. A 22(3), 280–288 (1977)

    Article  Google Scholar 

  46. Scott, L.L.: A conditions on Higman’s parameters. Not. Am. Math. Soc. 701, 20–45 (1973)

    Google Scholar 

  47. Seidel, J.J.: Strongly regular graphs with \((-1,1,0)\) adjacency matrix having eigenvalue \(3\). Linear Algebra Appl. 1, 281–298 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  48. Singleton, R.: On minimal graphs of maximum even girth. J. Combin. Theory 1, 306–332 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  49. Tarnanen, H.: Upper bounds on permutation codes via linear programming. Eur. J. Combin. 20, 101–114 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tutte, W.T.: Connectivity in Graphs. Toronto University Press, Toronto (1967)

    Google Scholar 

  51. Veblen, O., Maclagan-Wedderburn, J.H.: Non-Desarguesian and non-Pascalian geometries. Trans. Am. Math. Soc. 8(3), 379–388 (1907)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank Noga Alon, Eiichi Bannai, Tatsuro Ito, Jack Koolen, Hajime Tanaka, Sho Suda, and Masato Mimura for useful information and comments. The author is supported by JSPS KAKENHI Grant Numbers 25800011, 26400003.

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  1. Department of Mathematics Education, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi, 448-8542, Japan

    Hiroshi Nozaki

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  1. Hiroshi Nozaki
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Nozaki, H. Linear Programming Bounds for Regular Graphs. Graphs and Combinatorics 31, 1973–1984 (2015). https://doi.org/10.1007/s00373-015-1613-7

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