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A note on optimistic strongly regular graphs

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

Funds:  Supported by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China (61272008, 11271348, 10871189).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.03.001
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  • Author Bio:

    QIAO Zhi, male, born in 1990, PhD. Research field: Graph theory. E-mail：gesec@mail.ustc.edu.cn

  • Corresponding author: PAN Yongliang
  • Received Date: 17 November 2015
  • Accepted Date: 10 April 2016
  • Rev Recd Date: 10 April 2016
  • Publish Date: 30 March 2017
    Abstract

  • Abstract

    It is known that for a fixed integer α≥2, all but finitely many coconnected ones, the strongly regular graphs with smallest eigenvalue -α fall into two infinite families. Graham and Lovász raised the question of whether optimistic graphs exist and it was answered positively by Azarija. Here strongly regular graphs were classified with smallest eigenvalue -3, and the optimistic ones among them were determined.

    Abstract

    It is known that for a fixed integer α≥2, all but finitely many coconnected ones, the strongly regular graphs with smallest eigenvalue -α fall into two infinite families. Graham and Lovász raised the question of whether optimistic graphs exist and it was answered positively by Azarija. Here strongly regular graphs were classified with smallest eigenvalue -3, and the optimistic ones among them were determined.
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    • References(15)
  • [1]
    BOSE R C. Strongly regular graphs, partial geometries, and partially balanced designs[J]. Pacific J Math, 1963, 13: 389-419.
    [2]
    NEUMAIER A. Strongly regular graphs with least eigenvalue-m[J]. Arch Math, 1979, 33: 392-400.
    [3]
    GRAHAM R L, LOVSZ L. Distance matrix polynomials of trees[J]. Adv Math, 1978, 29: 60-88.
    [4]
    AZARIJA J. A short note on a short remark of Graham and Lovász[J]. Discrete Math, 2014, 315: 65-68.
    [5]
    BROUWER A E, COHEN A M, NEUMAIER A. Distance-Regular Graphs[M]. Berlin: Springer-Verlag, 1989.
    [6]
    BUSSEMAKER F C, HAEMERS W H, MATHON R, et al. A (49, 16, 3, 6) strongly regular graph does not exist[J]. European Journal of Combinatorics, 1989, 10: 413-418.
    [7]
    DEGRAER J. Isomorph-free exhaustive generation algorithms for association schemes[D]. Ghent, Belgium: Ghent University, 2007.
    [8]
    GODSIL C, Royle G. Algebraic Graph Theory[M]. New York: Springer-Verlag, 2001.
    [9]
    BONDARENKO A V, PRYMAK A, RADCHENKO D. Non-existence of (76,30,8,14) strongly regular graph and some structural tools[DB/OL]. arXiv:1410.6748.
    [10]
    HAEMERS W. There exists no (76, 21, 2, 7) strongly regular graph[C]// Finite Geometry and Combinatorics. Cambridge: Cambridge University Press, 1993: 175-176.
    [11]
    HAEMERS W H. Matrix techniques for strongly regular graphs and related geometries[EB/OL]. [2015-10-10]http://cage.ugent.be/fdc/intensivecourse2/haemers2.pdf.
    [12]
    SCOT L L JR. A condition on Higman’s parameters[J]. Notices Amer Math Soc, 1973, 701: 20-45.
    [13]
    BROUWER A E. Parameters of strongly regular graphs[EB/OL]. [2015-10-10]https://www.win.tue.nl/ aeb/graphs/srg/srgtab.html.
    [14]
    DELSARTE P, GOETHALS J M, SEIDEL J J. Bound for systems of lines and Jacobi polynomials[J]. Philips Res Rep, 1975, 30: 91-105.
    [15]
    WILBRINK H A, BROUWER A E. A (57,14,1) strongly regular graph does not exist[J]. Indag Math,1983, 86(1):117-121.)
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    [1]
    BOSE R C. Strongly regular graphs, partial geometries, and partially balanced designs[J]. Pacific J Math, 1963, 13: 389-419.
    [2]
    NEUMAIER A. Strongly regular graphs with least eigenvalue-m[J]. Arch Math, 1979, 33: 392-400.
    [3]
    GRAHAM R L, LOVSZ L. Distance matrix polynomials of trees[J]. Adv Math, 1978, 29: 60-88.
    [4]
    AZARIJA J. A short note on a short remark of Graham and Lovász[J]. Discrete Math, 2014, 315: 65-68.
    [5]
    BROUWER A E, COHEN A M, NEUMAIER A. Distance-Regular Graphs[M]. Berlin: Springer-Verlag, 1989.
    [6]
    BUSSEMAKER F C, HAEMERS W H, MATHON R, et al. A (49, 16, 3, 6) strongly regular graph does not exist[J]. European Journal of Combinatorics, 1989, 10: 413-418.
    [7]
    DEGRAER J. Isomorph-free exhaustive generation algorithms for association schemes[D]. Ghent, Belgium: Ghent University, 2007.
    [8]
    GODSIL C, Royle G. Algebraic Graph Theory[M]. New York: Springer-Verlag, 2001.
    [9]
    BONDARENKO A V, PRYMAK A, RADCHENKO D. Non-existence of (76,30,8,14) strongly regular graph and some structural tools[DB/OL]. arXiv:1410.6748.
    [10]
    HAEMERS W. There exists no (76, 21, 2, 7) strongly regular graph[C]// Finite Geometry and Combinatorics. Cambridge: Cambridge University Press, 1993: 175-176.
    [11]
    HAEMERS W H. Matrix techniques for strongly regular graphs and related geometries[EB/OL]. [2015-10-10]http://cage.ugent.be/fdc/intensivecourse2/haemers2.pdf.
    [12]
    SCOT L L JR. A condition on Higman’s parameters[J]. Notices Amer Math Soc, 1973, 701: 20-45.
    [13]
    BROUWER A E. Parameters of strongly regular graphs[EB/OL]. [2015-10-10]https://www.win.tue.nl/ aeb/graphs/srg/srgtab.html.
    [14]
    DELSARTE P, GOETHALS J M, SEIDEL J J. Bound for systems of lines and Jacobi polynomials[J]. Philips Res Rep, 1975, 30: 91-105.
    [15]
    WILBRINK H A, BROUWER A E. A (57,14,1) strongly regular graph does not exist[J]. Indag Math,1983, 86(1):117-121.)
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