  We propose a class of continuous-time quantum walk models on graphs induced
by a certain class of discrete-time quantum walk models with the parameter
$\epsilon\in [0,1]$. Here the graph treated in this paper can be applied both
finite and infinite cases. The induced continuous-time quantum walk is an
extended version of the (free) discrete-Schr\"odinger equation driven by the
normalized Laplacian: the element of the weighted Hermitian takes not only a
scalar value but also a matrix value depending on the underlying discrete-time
quantum walk. We show that each discrete-time quantum walk with an appropriate
setting of the parameter $\epsilon$ in the long time limit identifies with its
induced continuous-time quantum walk and give the running time for the
discrete-time to approximate the induced continuous-time quantum walk with a
small error $\delta$. We also investigate the detailed spectral information on
the induced continuous-time quantum walk.
