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In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachmentof one-length path periodically. This small modification to $\mathbb{Z}$provides localization of the quantum walk. The eigenspace causing thislocalization is generated by finite length round trip paths. We find that thelocalization is due to the eigenvalues of an underlying random walk. Moreoverwe find that the transience of the underlying random walk provides a slow downof the pseudo velocity of the induced quantum walk and a different limitdistribution from the Konno distribution.

The spreading behavior of quantum walks induced by drifted random walks on some magnifier graph