An Enumeration of Graphical Designs

Abstract

Let $Ψ(t,k)$ denote the set of pairs $(v,λ)$ for which there exists a graphical $t$-$(v,k,λ)$ design. Most results on graphical designs have gone to show the finiteness of $Ψ(t,k)$ when $t$ and $k$ satisfy certain conditions. The exact determination of $Ψ(t,k)$ for specified $t$ and $k$ is a hard problem and only $Ψ(2,3)$, $Ψ(2,4)$, $Ψ(3,4)$, $Ψ(4,5)$, and $Ψ(5,6)$ have been determined. In this paper, we determine completely the sets $Ψ(2,5)$ and $Ψ(3,5)$. 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