  We give graphical characterisation of the access structure to both classical
and quantum information encoded onto a multigraph defined for prime dimension
$q$, as well as explicit decoding operations for quantum secret sharing based
on graph state protocols. We give a lower bound on $k$ for the existence of a
$((k,n))_q$ scheme and prove, using probabilistic methods, that there exists
$\alpha$ such that a random multigraph has an accessing parameter $k\leq \alpha
n$ with high probability.
