  Let $E$ be a graph-directed set associated with a di-graph $G$. Let $\mu$ be
a Markov-type measure on $E$. Assuming a separation condition for $E$, we
determine the exact convergence order of the $L_r$-quantization error for
$\mu$. This result provides us with accurate information on the asymptotics of
the quantization error, especially when the quantization coefficient is
infinite.
