  The Grover walk is one of the most well-studied quantum walks on graphs. In
this paper, we investigate its periodicity to reveal the relationship between
the quantum walk and the underlying graph, focusing particularly on the
characterization of graphs exhibiting a periodic Grover walk. Graphs having a
periodic Grover walk with periods of $2, 3, 4$, and $5$ have previously been
characterized. It is expected that graphs exhibiting a periodic Grover walk
with odd period correspond to cycles with odd length. We address that problem
and are able to perfectly characterize the class of graphs exhibiting an
odd-periodic Grover walk by using a combinatorial method.
