  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.
