The Grover walk is one of well-studied quantum walks on graphs and its periodicity is investigated to reveal the relation between the quantum walk and the underlying graph. Especially, characterization of graphs exhibiting a periodic Grover walk is intensively studied. Yoshie has already characterized such graphs having a periodic Grover walk with periods $2, 3, 4$ and $5$. In the work, it is expected that the graphs exhibiting a periodic Grover walk with odd period are the cycles with odd length. In this paper, we address that problem and obtained the perfect answer, that is, we perfectly characterize the class of graphs exhibiting an odd-periodic Grover walk by a combinatorial method. More precisely, we solve the problem by analyzing the characteristic polynomial of a weighted adjacency matrix of the graph.
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