An Enumeration of Graphical Designs

Abstract

Let $\Psi (t,k)$ denote the set of pairs $(v,\lambda )$ for which there exists a graphical $t$-$(v,k,\lambda )$ design. Most results on graphical designs have gone to show the finiteness of $\Psi (t,k)$ when $t$ and $k$ satisfy certain conditions. The exact determination of $\Psi (t,k)$ for specified $t$ and $k$ is a hard problem and only $\Psi (2,3)$, $\Psi (2,4)$, $\Psi (3,4)$, $\Psi (4,5)$, and $\Psi (5,6)$ have been determined. In this paper, we determine completely the sets $\Psi (2,5)$ and $\Psi (3,5)$. As a result, we find more than 270000 inequivalent graphical designs, and more than 8000 new parameter sets for which there exists a graphical design. Prior to this, graphical designs are known for only 574 parameter sets.

16 pages