The Edge-Connectivity of Strongly 3-Walk-Regular Graphs

  • Rongquan Feng
  • Wenqian ZhangEmail author
Original Paper


E.R. van Dam and G.R. Omidi generalized the concept of strongly regular graphs as follows. If for any two vertices the number of \(\ell \)-walks (walks of length \(\ell \)) from one vertex to the other is the same which depends only on whether the two vertices are the same, adjacent or non-adjacent, then G is called a strongly \(\ell \)-walk-regular graph. The existence of strongly \(\ell \)-walk-regular graphs which are not strongly 3-walk-regular graphs is unknown. In this paper, we prove that the edge-connectivity of a connected strongly 3-walk-regular graph G of degree \(k\ge 3\) is equal to k. Moreover, if G is not the graph formed by adding a perfect matching between two copies of \(K_{4}\), then each edge cut set of size k is precisely the set of edges incident with a vertex of G. For a regular graph G in general, we also give a sufficient and tight condition such that G is 1-extendable.


Edge-connectivity Eigenvalue Strongly 3-walk-regular graph Perfect matching 1-Extendable 

Mathematics Subject Classification

05C40 05C50 05C70 



The authors are grateful for the useful comments which improve the paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 61370187 and 11731002).


  1. 1.
    Anderson, I.: Perfect matchings of a graph. J. Combin. Theory 10, 183–186 (1971)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)CrossRefGoogle Scholar
  3. 3.
    Brouwer, A.E., Mesner, D.M.: The connectivity of strongly regular graphs. Eur. J. Combin. 6, 215–216 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cioabă, 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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cioabă, S.M., Koolen, J.H., Li, W.: Disconnecting strongly regular graphs. Eur. J. Combin. 38, 1–11 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cvetković, D., Rowlinson, P., Simić, S.: Eigenspace of Graphs. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  7. 7.
    Duval, A.M.: A directed graph version of strongly regular graphs. J. Combin. Theory, Ser. A 47, 71–100 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    van Dam, E.R.: Regular graphs with four eigenvalues. Linear Algebra Appl. 226–228, 139–162 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    van Dam, E.R., Omidi, G.R.: Strongly walk-regular graphs. J. Combin. Theory, Ser. A 120, 803–810 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    van Dam, E.R., Omidi, G.R.: Directed strongly walk-regular graphs. J. Algebr. Combin. 120, 1–17 (2015)zbMATHGoogle Scholar
  11. 11.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, New York (2001)CrossRefGoogle Scholar
  12. 12.
    Haemers, W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 226, 593–616 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lovász, L., Plummer, M. D.: Matching Theory, Ann. Discrete Math (1986)Google Scholar
  14. 14.
    Tutte, W.T.: The factorization of linear graphs. J. Lond. Math. Soc. 2, 107–111 (1947)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhang, W.: The cyclic edge-connectivity of strongly regular graphs. Graphs Combin. 35, 779–785 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China

Personalised recommendations