The quantization for Markov-type measures on a class of ratio-specified graph directed fractals

We study the asymptotic quantization error of order $r$ for Markov-type measures $\mu$ on a class of ratio-specified graph directed fractals. We show that the quantization dimension of $\mu$ exists and determine its exact value $s_{r}$ in terms of spectral radius of a related matrix. We prove that the $s_{r}$-dimensional lower quantization coefficient of $\mu$ is always positive. Moreover, inspired by Mauldin-Williams's work on the Hausdorff measure of graph directed fractals, we establish a necessary and sufficient condition for the $s_{r}$-dimensional upper quantization coefficient of $\mu$ to be finite.