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The Limit Theorem with Respect to the Matrices on Non-backtracking Paths of a Graph

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Abstract

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the kth moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the \(p^m\)th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

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Notes

  1. Huang’s \(T_k(x)\) in [17, (8)] is not the Chebyshev polynomial of the first kind, and his \(T_k(x)\) is equal to our \(2\,T_k(x/2)\) as in [17, §3].

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Acknowledgements

The authors would like to thank the anonymous referee for careful reading and suggesting improvements of some statements. Takehiro Hasegawa was partially supported by JSPS KAKENHI (grant numbers 19K03400 and 22K03246). Hayato Saigo was partially supported by JSPS KAKENHI (grant numbers 19K03608 and 22K03405) and by Research Origin for Dressed Photon. Seiken Saito was partially supported by JSPS KAKENHI (grant numbers 19K03608 and 22K03405) and by Research Origin for Dressed Photon. Shingo Sugiyama was partially supported by JSPS KAKENHI (grant number 20K14298).

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Authors and Affiliations

  1. Department of Education, Shiga University, Otsu, Shiga, 520-0862, Japan

    Takehiro Hasegawa

  2. Math Research Institute Calc for Industry, Minami, Hiroshima, 732-0816, Japan

    Takashi Komatsu

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

    Norio Konno

  4. Nagahama Institute of Bio-Science and Technology, 1266, Tamura, Nagahama, 526-0829, Japan

    Hayato Saigo

  5. Division of Liberal Arts, Center for Promotion of Higher Education, Kogakuin University, 2665-1 Nakano, Hachioji, Tokyo, 192-0015, Japan

    Seiken Saito

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

    Iwao Sato

  7. Department of Mathematics, College of Science and Technology, Nihon University, Suruga-Dai, Kanda, Chiyoda, Tokyo, 101-8308, Japan

    Shingo Sugiyama

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  1. Takehiro Hasegawa
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  2. Takashi Komatsu
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Correspondence to Seiken Saito.

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Hasegawa, T., Komatsu, T., Konno, N. et al. The Limit Theorem with Respect to the Matrices on Non-backtracking Paths of a Graph. Ann. Comb. 27, 249–268 (2023). https://doi.org/10.1007/s00026-022-00617-z

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  • DOI: https://doi.org/10.1007/s00026-022-00617-z

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