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Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

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Abstract

For a graph with at most one loop at each vertex, we define a discrete-time quaternionic quantum walk on the graph, which can be viewed as a quaternionic extension of the Grover walk on the graph. We derive the unitary condition for the transition matrix of the quaternionic Grover walk, and discuss the relationship between the right spectra of the transition matrices and zeta functions of graphs.

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Acknowledgements

N. Konno is partially supported by the Grant-in-Aid for Scientific Research (Challenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No. 15K13443). H. Mitsuhashi and I. Sato are partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 16K05249 and No. 15K04985, respectively). We thank K. Tamano, Y. Ide, and O. Kada for helpful comments at the stage of the revision process. We would also like to thank the referee for careful reading and for fruitful suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama, Kanagawa, 240-8501, Japan

    Norio Konno

  2. Department of Applied Informatics, Faculty of Science and Engineering, Hosei University, Koganei, Tokyo, 184-8584, Japan

    Hideo Mitsuhashi

  3. Oyama National College of Technology, Oyama, Tochigi, 323-0806, Japan

    Iwao Sato

Authors
  1. Norio Konno
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  2. Hideo Mitsuhashi
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  3. Iwao Sato
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Correspondence to Hideo Mitsuhashi.

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Konno, N., Mitsuhashi, H. & Sato, I. Quaternionic Grover Walks and Zeta Functions of Graphs with Loops. Graphs and Combinatorics 33, 1419–1432 (2017). https://doi.org/10.1007/s00373-017-1785-4

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  • DOI: https://doi.org/10.1007/s00373-017-1785-4

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