Skip to main content
SpringerLink
Log in

Locally triangular graphs and normal quotients of the n-cube

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

For an integer \(n\ge 2\), the triangular graph has vertex set the 2-subsets of \(\{1,\ldots ,n\}\) and edge set the pairs of 2-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-path lies in a unique quadrangle. We refine this result and provide a characterisation of connected locally triangular graphs as halved graphs of normal quotients of n-cubes. To do so, we study a parameter that generalises the concept of minimum distance for a binary linear code to arbitrary automorphism groups of the n-cube.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bamberg, J., Devillers, A., Fawcett, J.B., Praeger, C.E.: Locally triangular graphs and rectagraphs with symmetry. J. Comb. Theory Ser. A 133, 1–28 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brouwer, A.E.: Classification of small \((0,2)\)-graphs. J. Comb. Theory Ser. A 113, 1636–1645 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  5. Jurišić, A., Koolen, J.: 1-homogeneous graphs with cocktail party \(\mu \)-graphs. J. Algebraic Comb. 18, 79–98 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Makhnev, A.A.: On the graphs with \(\mu \)-subgraphs isomorphic to \(K_{u\times 2}\). Proc. Steklov Inst. Math. Suppl. 2, S169–S178 (2001). Translated from Trudy Instituta Matematiki UrO RAN, Vol.7, No. 2 (2001)

  7. Matsumoto, M.: On the classification of locally Hamming distance-regular graphs. RIMS Kôkyûroku 768, 50–61 (1991). http://www.kurims.kyoto-u.ac.jp/kyodo/kokyuroku/contents/pdf/0768-07.pdf

  8. Neumaier, A.: Rectagraphs, diagrams, and Suzuki’s sporadic simple group. Ann. Discrete Math. 15, 305–318 (1982)

    MathSciNet  MATH  Google Scholar 

  9. Perkel, M.: On Finite Groups Acting on Polygonal Graphs. Ph.D. thesis, University of Michigan (1977)

Download references

Author information

Authors and Affiliations

  1. Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Joanna B. Fawcett

Authors
  1. Joanna B. Fawcett
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joanna B. Fawcett.

Additional information

The author was supported by the Australian Research Council Discovery Project Grant DP130100106 and thanks the referees for helpful observations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fawcett, J.B. Locally triangular graphs and normal quotients of the n-cube. J Algebr Comb 44, 119–130 (2016). https://doi.org/10.1007/s10801-015-0659-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-015-0659-1

Keywords

Search

Navigation