Graphical t-designs via polynomial Kramer-Mesner matrices

Anton Betten, Mikhail Klin, Reinhard Laue, Alfred Wassermann

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


Kramer-Mesner matrices have been used as a powerful tool to construct t-designs. In this paper we construct Kramer-Mesner matrices for fixed values of k and t in which the entries are polynomials in n the number of vertices of the underlying graph. From this we obtain an elementary proof that with a few exceptions S[2]n is a maximal subgroup of S(n2) or A(n2). We also show that there are only finitely many graphical incomplete t-(υ, k, λ) designs for fixed values of 2≤t and k at least in the cases k = t + 1, t = 2, and 2≤t<k≤6. All graphical t-designs are determined by the program DISCRETA3 for various small parameters. Most parameter sets are new for graphical designs, some also for general simple t-designs. The largest value of t for which graphical designs were found is t = 5. Some of the smaller designs which are block transitive are drawn as graphs.

Original languageEnglish
Pages (from-to)83-109
Number of pages27
JournalDiscrete Mathematics
StatePublished - 28 Feb 1999


  • Graphical t-designs
  • Maximal subgroups of symmetric groups
  • Polynomial kramer-Mesner matrices

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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