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Adjacency and transition matrices related to random walks on graphs

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Abstract

A pointed graph \((\Gamma ,v_0)\) induces a family of transition matrices in Wildberger’s construction of a hermitian hypergroup using a random walk on \(\Gamma \) starting from \(v_0\). In this study, we propose a necessary condition for producing a hermitian hypergroup considering a weaker condition than the distance-regularity for \((\Gamma ,v_0)\). Furthermore, we show that the condition obtained connects the transition and adjacency matrices associated with \(\Gamma \).

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Acknowledgements

The authors thank Kenta Endo and Ippei Mimura for their helpful comments. This work was supported by JSPS KAKENHI Grant-in-Aid for Research Activity Start-up (No. 19K23403). The authors would also like to thank MARUZEN-YUSHODO Co., Ltd. (https://kw.maruzen.co.jp/kousei-honyaku/) for the English language editing.

Funding

This work was supported by JSPS (Japan Society for the Promotion of Science) KAKENHI Grant-in-Aid for Research Activity Start-up (No. 19K23403).

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Authors and Affiliations

  1. Graduate school of mathematics, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8602, Japan

    Tomohiro Ikkai

  2. Department of Mathematics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan

    Hiromichi Ohno

  3. Department of Mathematics, Hokkaido University, North 10 West 8, Kita-ku, Sapporo, 060-0810, Japan

    Yusuke Sawada

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  1. Tomohiro Ikkai
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  2. Hiromichi Ohno
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  3. Yusuke Sawada
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Correspondence to Tomohiro Ikkai.

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Ikkai, T., Ohno, H. & Sawada, Y. Adjacency and transition matrices related to random walks on graphs. J Algebr Comb 56, 249–267 (2022). https://doi.org/10.1007/s10801-021-01107-w

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