Abstract
We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product, which is regarded as the Euler product. This is a quaternionic extension of the square of the Ihara zeta function. We show that the new zeta function can be expressed as the exponential of a generating function and that it has two Study determinant expressions, which are crucial for the theory of zeta functions of graphs.
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Acknowledgments
We are grateful to H. Morita for some valuable comments on this work. We would also like to thank the referees for careful reading of the paper and valuable suggestions and comments which helped to improve the manuscript.
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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). I. Sato is partially supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science (Grant No. 15K04985).
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Konno, N., Mitsuhashi, H. & Sato, I. The quaternionic weighted zeta function of a graph. J Algebr Comb 44, 729–755 (2016). https://doi.org/10.1007/s10801-016-0686-6
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DOI: https://doi.org/10.1007/s10801-016-0686-6