  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
$\mathbb{Z}$. 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 {\it localization} in the Grover
walks, in that 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
$(-1/\sqrt{10},1/\sqrt{10})$ and $(-2/7,2/7)$, respectively.
