Graphs and Combinatorics

, Volume 35, Issue 3, pp 779–785 | Cite as

The Cyclic Edge-Connectivity of Strongly Regular Graphs

  • Wenqian ZhangEmail author
Original Paper


Let G be a connected graph. An edge cut set M of G is a cyclic edge cut set if there are at least two components of \(G-M\) which contain a cycle. The cyclic edge-connectivity of G is the minimum cardinality of a cyclic edge cut set (if exists) of G. In this paper, we show that the cyclic edge-connectivity of a connected strongly regular graph G (not \(K_{3,3}\)) of degree \(k\ge 3\) with girth c is equal to \((k-2)c\), where \(c=3, 4\) or 5. Moreover, if G is not the triangular graph srg-(10, 6, 3, 4), the complete multi-partite graph \(K_{2,2,2,2}\) or the lattice graph srg-(16, 6, 2, 2), then each cyclic edge cut set of size \((k-2)c\) is precisely the set of edges incident with a cycle of length c in G.


Strongly regular graph Eigenvalue Edge cut set Cyclic edge-connectivity 

Mathematics Subject Classification

05C40 05C50 05C75 



The author would like to thank the editors and the anonymous referees for their helpful comments on improving the representation of the paper.


  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, Macmillan, London. Elsevier, New York (1976)CrossRefGoogle Scholar
  2. 2.
    Brouwer, A.E.: Parameters of strongly regular graphs, list available at
  3. 3.
    Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Brouwer, A.E., Haemers, W.H.: Eigenvalues and perfect matchings. Linear Algebra Appl. 395, 155–162 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brouwer, A.E., Koolen, J.H.: The vertex-connectivity of distance-regular graphs. Eur. J. Combin. 30, 668–673 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brouwer, A.E., Mesner, D.M.: The connectivity of strongly regular graphs. Eur. J. Combin. 6, 215–216 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cioabǎ, S.M., Kim, K., Koolen, J.H.: On a conjecture of Brouwer involving the connectivity of strongly regular graphs. J. Combin. Theory Ser. A 119, 904–922 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cioabǎ, S.M., Koolen, J.H., Li, W.: Disconnecting strongly regular graphs. Eur. J. Combin. 38, 1–11 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cioabǎ, S.M., Li, W.: The extendability of matchings in strongly regular graphs. Electron. J. Combin. 21(2), 23 (2014). (Paper 2.34)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6, 363–388 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Godsil, C., Royle, G.: Algebraic Graph Theory (Graduate Texts in Mathematics, vol. 207). Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Haemers, W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 227–228, 593–616 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    McCuaig, W.: Edge-reductions in cyclically \(k\)-connected cubic graphs. J. Combin. Theory Ser. B 56, 16–44 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Neumaier, A.: Strongly regular graphs with smallest eigenvalue \(-m\). Arch. Math. 33(4), 392–400 (1979-80)Google Scholar
  15. 15.
    Seidel, J.J.: Strongly regular graphs with \((-1,1,0)\) adjacency matrix having eigenvalue \(3\). Linear Algebra Appl. 1, 281–298 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tutte, W.T.: The factorization of linear graphs. J. Lond. Math. Soc. 22, 107–111 (1947)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesPeking UniversityBeijingChina

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