Graphs and Combinatorics

, Volume 35, Issue 6, pp 1273–1291 | Cite as

On the Integrability of Strongly Regular Graphs

  • Jack H. Koolen
  • Masood Ur Rehman
  • Qianqian YangEmail author
Original Paper


Koolen et al. showed that if a connected graph with smallest eigenvalue at least \(-3\) has large minimal valency, then it is 2-integrable. In this paper, we will prove that a lower bound for the minimal valency is 166.


Strongly regular graph Lattice s-Integrability 

Mathematics Subject Classification

05C50 05E30 11H99 



The first author J.H. Koolen is partially supported by the National Natural Science Foundation of China (Grant No. 11471009 and Grant No. 11671376) and Anhui Initiative in Quantum Information Technologies (No. AHY150000). The second author M.U. Rehman is supported by the Chinese Scholarship Council at USTC, China. The corresponding author Q. Yang is partially supported by the Chinese Scholarship Council (No. 201806340049). We also would like to thank Akihiro Munemasa for pointing out that one can extend the complement of the McLaughlin graph to obtain a slightly better lower bound.


  1. 1.
    Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)CrossRefGoogle Scholar
  2. 2.
    Brouwer, A.E., Haemers, W.H.: Structure and uniqueness of the \((81,20,1,6)\) strongly regular graph. Discret. Math. 106/107, 77–82 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brouwer, A.E., van Lint, J.H.: Strongly regular graphs and partial geometries. Enumer. Des. 85, 122 (1982)Google Scholar
  4. 4.
    Cameron, P.J., Goethals, J.M., Seidel, J.J.: Strongly regular graphs having strongly regular subconstituents. J. Algebra 552, 57–280 (1978)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cameron, P.J., Goethals, J.M., Seidel, J.J., Shult, E.E.: Line graphs, root systems and elliptic geometry. J. Algebra 43, 305–327 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cameron, P.J., van Lint, J.H.: Graphs, Codes and Designs. Cambridge University Press, Cambridge (1980)CrossRefGoogle Scholar
  7. 7.
    Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Springer, New York (1988)CrossRefGoogle Scholar
  8. 8.
    Conway, J.H., Sloane, N.J.A.: Low-dimensional lattices. V. Integral coordinates for integral lattices. Proc. R. Soc. Lond. Ser. A 426, 211–232 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, vi+-97 (1973)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Deza, M., Grishukhin, V.P., Laurent, M.: Hypermetrics in Geometry of Numbers (DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 20), pp. 1–109. American Mathematical Socirty, Providence (1995)Google Scholar
  11. 11.
    Fan, C., Schwenk, A.J.: Structure of the Hoffman–Singleton graph. Congr. Numer. 94, 3–8 (1993)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gewirtz, A.: The uniquence of \(g(2,2,10,56)\). Trans. N Y Acad. Sci. 31, 656–675 (1969)CrossRefGoogle Scholar
  13. 13.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)CrossRefGoogle Scholar
  14. 14.
    Goethals, J.M., Seidel, J.J.: The regular two-graph on \(276\) vertices. Discret. Math. 12, 143–158 (1975)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Haemers, W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 226–228, 593–616 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hafner, P.R.: The Hoffman–Singleton graph and its automorphisms. J. Algebraic Combin. 18, 7–12 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hoffman, A.J.: On graphs whose least eigenvalue exceeds \(-1-\sqrt{2}\). Linear Algebra Appl. 16, 153–165 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hoffman, A.J., Singleton, R.R.: On Moore graphs with diameters 2 and 3. IBM J. Res. Dev. 4, 497–504 (1960)MathSciNetCrossRefGoogle Scholar
  19. 19.
    James, L.O.: A combinatorial proof that the Moore \((7,2)\) graph is unique. Utilitas Math. 5, 79–84 (1974)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Koolen, J.H., Yang, J.Y., Yang, Q.: On graphs with smallest eigenvalue at least \(-3\) and their lattices. Adv. Math. 338, 847–864 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Neumaier, A.: Strongly regular graphs with smallest eigenvalue \(-m\). Arch. Math. (Basel), 33 392–400 (1979/80)Google Scholar
  22. 22.
    Nyatate, S.M., Pawale, R.M., Shrikhande, M.S.: Characterization of quasi-symmetric designs with eigenvalues of their block graphs. Austral. J. Combin. 68, 62–70 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Pawale, R.M.: Inequalities and bounds for quasi-symmetric 3-designs. J. Combin. Theory Ser A. 60(2), 159–167 (1992)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Payne, S.E., Thas, J.A.: Finite generalized quadrangles. In: EMS Series of Lectures in Mathematics, second ed. European Mathematical Society (EMS), Zürich (2009)Google Scholar
  25. 25.
    Ramsey, F.P.: On a problem of formal logic. Proc. Lond. Math. Soc. 48, 264–286 (1930)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Seidel, J.J.: Strongly regular graphs with \((-1,1,0)\) adjacency matrix having eigenvalue \(3\). Linear Algebra Appl. 1, 281–298 (1968)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Stinson, D.R.: Combinatorial Designs: Construction and Analysis. Springer, New York (2004)zbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Wen-Tsun Wu Key Laboratory of CAS, School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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