An Enumeration of Graphical Designs

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. c © John Wiley & Sons, Inc.