Journal of Algebraic Combinatorics

, Volume 44, Issue 1, pp 119–130 | Cite as

Locally triangular graphs and normal quotients of the n-cube

  • Joanna B. Fawcett


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.


Locally triangular graph Rectagraph Normal quotient n-cube Semibiplane 


  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brouwer, A.E.: Classification of small \((0,2)\)-graphs. J. Comb. Theory Ser. A 113, 1636–1645 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  5. 5.
    Jurišić, A., Koolen, J.: 1-homogeneous graphs with cocktail party \(\mu \)-graphs. J. Algebraic Comb. 18, 79–98 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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)Google Scholar
  7. 7.
    Matsumoto, M.: On the classification of locally Hamming distance-regular graphs. RIMS Kôkyûroku 768, 50–61 (1991).
  8. 8.
    Neumaier, A.: Rectagraphs, diagrams, and Suzuki’s sporadic simple group. Ann. Discrete Math. 15, 305–318 (1982)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Perkel, M.: On Finite Groups Acting on Polygonal Graphs. Ph.D. thesis, University of Michigan (1977)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Centre for the Mathematics of Symmetry and ComputationThe University of Western AustraliaCrawleyAustralia

Personalised recommendations