Trapping and spreading properties of quantum walk in homological structure

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