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Monotonic Normalized Heat Diffusion for Regular Bipartite Graphs with Four Eigenvalues

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Abstract

Let \(X=(V, E)\) be a finite regular graph and \(H_t(u, v), \, u, v \in V\), the heat kernel on X. We prove that, if the graph X is bipartite and has four distinct Laplacian eigenvalues, the ratio \(H_t(u, v)/H_t(u, u), \, u, v \in V,\) is monotonically non-decreasing as a function of t. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design.

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Acknowledgements

The second-named author would like to thank Professor Ryokichi Tanaka for informing him of the results in Regev–Shinkar [9] and Price [8] when he visited Tohoku University in June 2017, and for giving him valuable comments. We also thank the anonymous referees for valuable comments.

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Authors and Affiliations

  1. Garduate School of Science and Engineering, Ritsumeikan University, 1-1-1, Noji-Higashi, Kusatsu, 525-8529, Japan

    Tasuku Kubo

  2. Department of Mathematics, Faculty of Education, Shizuoka University, 836, Ohya, Suruga-ku, Shizuoka, 422-8529, Japan

    Ryuya Namba

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  1. Tasuku Kubo
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  2. Ryuya Namba
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Correspondence to Ryuya Namba.

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This work is supported by JSPS KAKENHI Grant Number 19K23410.

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Kubo, T., Namba, R. Monotonic Normalized Heat Diffusion for Regular Bipartite Graphs with Four Eigenvalues. Graphs and Combinatorics 38, 22 (2022). https://doi.org/10.1007/s00373-021-02424-4

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  • DOI: https://doi.org/10.1007/s00373-021-02424-4

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