TY - JOUR

T1 - Graphical t-designs via polynomial Kramer-Mesner matrices

AU - Betten, Anton

AU - Klin, Mikhail

AU - Laue, Reinhard

AU - Wassermann, Alfred

N1 - Funding Information:
Consider the action of the symmetric group S,, on the set V= (x) where X= {1,2, .... n}. This defines an embedding of S, into S(g) with image group S, f2\]. Any subset K of V can be considered as a labelled graph with edge set K and vertex set X. The orbits of S,I 2\] on 2 v are just the isomorphism classes of graphs and thus such an orbit * Corresponding author. E-mail: laue@uni-bayreuth.de. 1 Supported by the Deutsche Forschungsgemeinschafl, Ke 201/17-1. 2 Partially supported by the Israeli Ministry of Absorption and by the research grant No 6782-1-95 of the Israeli Ministry of Science. 3 A. Betten, R. Laue, A. Wassermann: DISCRETA, a program system for the construction of t-Designs with a prescribed automorphism group. University of Bayreuth, I998. http://www.mathe2.uni-bayreuth.de/ betten/DISCRETA/Index.html.

PY - 1999/2/28

Y1 - 1999/2/28

N2 - 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≤t3 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.

AB - 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≤t3 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.

KW - Graphical t-designs

KW - Maximal subgroups of symmetric groups

KW - Polynomial kramer-Mesner matrices

UR - http://www.scopus.com/inward/record.url?scp=18544409723&partnerID=8YFLogxK

U2 - 10.1016/s0012-365x(99)90045-6

DO - 10.1016/s0012-365x(99)90045-6

M3 - Article

AN - SCOPUS:18544409723

SN - 0012-365X

VL - 197-198

SP - 83

EP - 109

JO - Discrete Mathematics

JF - Discrete Mathematics

ER -