Related papers: The discrete-time Grover walk on star graphs with …

This paper has been withdrawn by the author, due to a crucial error in page 5.

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…

This paper has been withdrawn by the authors; its contents are superseded by that of hep-th/0105305 and hep-th/0212275.

This paper has been withdrawn by the author because there is a gap in Lemma 9.

This paper has been withdrawn by the author, due a critical mistake on page 3.

This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089.

This paper has been withdrawn by the authors due to several major changes in concepts and results. New version will follow briefly as a new e-print.

This paper has been withdrawn by the author due to the presented idea is wrong.

This paper has been withdrawn by the author due to a missing figure

A modified version of this paper is under process and with a new title and abstract. Hence, this version of the article is completely withdrawn.

The above mentioned paper is withdrawn.

This paper has been withdrawn by the author.

The authors have withdrawn this paper.

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This paper has been withdrawn by the author.

This paper has been withdrawn

The paper has been withdrawn.

This paper has been withdrawn by the author

This paper has been withdrawn

This paper has been withdrawn by the author. It will be replaced, substantially modified, by sections of the author's completed PhD thesis.