Graphs and Combinatorics

, Volume 33, Issue 6, pp 1419–1432 | Cite as

Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

  • Norio Konno
  • Hideo Mitsuhashi
  • Iwao Sato
Original Paper


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.


Quantum walk Ihara zeta function Quaternion 



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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Copyright information

© Springer Japan 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of EngineeringYokohama National UniversityHodogaya, YokohamaJapan
  2. 2.Department of Applied Informatics, Faculty of Science and EngineeringHosei UniversityKoganeiJapan
  3. 3.Oyama National College of TechnologyOyamaJapan

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