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We attempt to extract a homological structure of two kinds of graphs by theGrover walk. The first one consists of a cycle and two semi-infinite lines andthe second one is assembled by a periodic embedding of the cycles in$\mathbb{Z}$. We show that both of them have essentially the same eigenvaluesinduced by the existence of cycles in the infinite graphs. The eigenspace ofthe homological structure appears as so called {\it localization} in the Groverwalks, in that the walk is partially trapped by the homological structure. Onthe other hand, the difference of the absolutely continuous part of spectrumbetween them provides different behaviors. We characterize the behaviors by thedensity functions in the weak convergence theorem: the first one is the deltameasure at the bottom while the second one is expressed by two kinds ofcontinuous functions which have different finite supports$(-1/\sqrt{10},1/\sqrt{10})$ and $(-2/7,2/7)$, respectively.

Trapping and spreading properties of quantum walk in homological structure