Israel Journal of Mathematics

, Volume 196, Issue 1, pp 295–319 | Cite as

Clique complexes and graph powers

  • Michał Adamaszek


We study the behaviour of clique complexes of graphs under the operation of taking graph powers. As an example we compute the clique complexes of powers of cycles, or, in other words, the independence complexes of circular complete graphs.


Simplicial Complex Homotopy Type Weak Equivalence Homotopy Equivalent Circulant Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. A. Barmak, Star clusters in independence complexes of graphs, arxiv/1007.0418.Google Scholar
  2. [2]
    C. Bachoc, A. Pêcher and A. Thiéry, On the theta number of powers of cycle graphs, arxiv/1103.0444.Google Scholar
  3. [3]
    A. Björner, Topological methods, in Handbook of Combinatorics, (R. Graham, M. Grötschel and L. Lovász, eds.), North-Holland, Amsterdam, 1995, pp. 1819–1872.Google Scholar
  4. [4]
    R. Boulet, E. Fieux and B. Jouve, Simplicial simple-homotopy of flag complexes in terms of graphs, European Journal of Combinatorics 31 (2010), 161–176.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    B. Braun, Independence complexes of stable Kneser graphs, Electronic Journal of Combinatorics 18 (2011), paper 118.Google Scholar
  6. [6]
    A. E. Brouwer, P. Duchet and A. Schrijver, Graphs whose neighbourhoods have no special cycles, Discrete Mathematics 47 (1983), 177–182.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Brown and R. Hoshino, Independence polynomials of circulants with an application to music, Discrete Mathematics 309 (2009), 2292–2304.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Dochtermann, The universality of Hom complexes of graphs, Combinatorica 29 (2009), 433–448.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    R. Forman, Morse theory for cell complexes, Advances in Mathematics 134 (1998), 90–145.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Forman, A user’s guide to discrete Morse theory, Séminaire Lotharingien de Combinatoire 48 (2002), article B48c.MathSciNetGoogle Scholar
  11. [11]
    R. Hoshino, Independence polynomials of circulant graphs, PhD. Thesis, Dalhouse University, 2007.Google Scholar
  12. [12]
    D. Kozlov, Complexes of directed trees, Journal of Combinatorial Theory. Series A 88 (1999), 112–122.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Kozlov, Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics, Vol. 21, Springer-Verlag, Berlin-Heidelberg, 2008.CrossRefzbMATHGoogle Scholar
  14. [14]
    F. Larrión, M. A. Pizaña and R. Villarroel-Flores, The fundamental group of the clique graph, European Journal of Combinatorics 30 (2009), 288–294.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. P. May, Weak equivalences and quasifibrations, in Groups of Self-equivalences and Related Topics (R. Piccinini, ed.), Lecture Notes in Mathematics, Vol. 1425, Springer-Verlag, Berlin, 1990, pp. 91–101.CrossRefGoogle Scholar
  16. [16]
    M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Mathematical Journal 33 (1966), 465–474.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    R. Motwani and M. Sudan, Computing roots of graphs is hard, Discrete Applied Mathematics 54 (1994), 81–88.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Muzychuk, A solution of the isomorphism problem for circulant graphs, Proceedings of the London Mathematical Society 88 (2004), 1–41.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2013

Authors and Affiliations

  1. 1.Mathematics Institute and DIMAPUniversity of WarwickCoventryUK

Personalised recommendations