Graphical t-designs via polynomial Kramer-Mesner matrices

Kramer-Mesner matrices have been used as a powerful tool to construct t-designs. In this paper we construct Kramer-Mesner matrices for xed 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 (n 2) or A (n 2). We also show that there are only nitely many graphical incomplete t-(v; k;) designs for xed 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.