Graphical t-designs via polynomial Kramer-Mesner matrices Original Research Article

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 Sn[2] is a maximal subgroup of Sn2 or An2. We also show that there are only finitely many graphical incomplete t-(v,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 DISCRETA 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.