An Enumeration of Graphical Designs

TL;DR

This paper completely determines the sets of parameters for certain graphical designs, significantly expanding known examples from 574 to over 278,000, and introduces new parameter sets where such designs exist.

Contribution

It fully characterizes the sets Psi(2,5) and Psi(3,5), vastly increasing known graphical design parameters and numbers.

Findings

Over 270,000 inequivalent graphical designs identified

More than 8,000 new parameter sets established for graphical designs

Expanded the total known parameter sets from 574 to over 278,000

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.