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Distance-transitive dihedrants

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Abstract

The main result of this article is a classification of distance-transitive Cayley graphs on dihedral groups. We show that a Cayley graph X on a dihedral group is distance-transitive if and only if X is isomorphic to one of the following graphs: the complete graph K 2n ; a complete multipartite graph K t×m with t anticliques of size m, where t m is even; the complete bipartite graph without 1-factor K n,n  − nK 2; the cycle C 2n ; the incidence or the non-incidence graph of the projective geometry PG d-1(d,q), d ≥ 2; the incidence or the non-incidence graph of a symmetric design on 11 vertices.

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References

  1. Beth T, Jungnickel D, Lenz H (1999) Design theory, vol 1, Second Edition, Encyclopedia of Mathematics and its Applications 69. Cambridge University Press, Cambridge

    Google Scholar 

  2. Brouwer AE, Cohen AM, Neumaier A (1998) Distance-regular graphs. Springer-Verlag, New York

    MATH  Google Scholar 

  3. Jungnickel D, Pott A (1996) Abelian difference sets. In: Coulbourn CJ, Dinitz JH (eds) The CRC hanbook of combinatorial designs. CRC Press, New York

    Google Scholar 

  4. Kantor W (1985) Classification of 2-transitive symmetric designs. Graphs Combin. 1:165–166

    MATH  MathSciNet  Google Scholar 

  5. Miklavič Š, Potočnik P (2003) Distance-regular circulants. Eur J Combin 24:777–784

    Article  Google Scholar 

  6. Miklavič Š, Potočnik P (xxxx) Distance-regular cayley graphs on dihedral groups accepted in J Combin Theory Ser B

  7. Paley REAC (1933) On orthogonal matrices. J Math Phys MIT 12:311–320

    MATH  Google Scholar 

  8. Shmidt B (2002) Characters and cyclotomic fields in finite geometry. Lecture Notes in Mathematics 1797. Springer-Verlag, New York

    MATH  Google Scholar 

  9. Singer J (1938) A theorem in finite projective geometry and some applications to number theory. Trans Am Math Soc 43:377–385

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Education, University of Primorska, Cankarjeva 5, 6000, Koper, Slovenia

    Štefko Miklavič

  2. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    Primož Potočnik

Authors
  1. Štefko Miklavič
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  2. Primož Potočnik
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Correspondence to Štefko Miklavič.

Additional information

Communicated by C. E. Praeger.

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Miklavič, Š., Potočnik, P. Distance-transitive dihedrants. Des Codes Crypt 41, 185–193 (2006). https://doi.org/10.1007/s10623-006-9008-7

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  • DOI: https://doi.org/10.1007/s10623-006-9008-7

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