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Sedentariness in quantum walks

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Abstract

We formalize the notion of a sedentary vertex and present a relaxation of the concept of a sedentary family of graphs introduced by Godsil (Linear Algebra Appl 614:356–375, 2021. https://doi.org/10.1016/j.laa.2020.08.027). We provide sufficient conditions for a given vertex in a graph to exhibit sedentariness. We also show that a vertex with at least two twins (vertices that share the same neighbours) is sedentary. We prove that there are infinitely many graphs containing strongly cospectral vertices that are sedentary, which reveals that, even though strong cospectrality is a necessary condition for pretty good state transfer, there are strongly cospectral vertices which resist high probability state transfer to other vertices. Moreover, we derive results about sedentariness in products of graphs which allow us to construct new sedentary families, such as Cartesian powers of complete graphs and stars.

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Acknowledgements

I thank the University of Manitoba Faculty of Science and Faculty of Graduate Studies for the support. I thank Steve Kirkland, Sarah Plosker, Chris Godsil and Cristino Tamon for the helpful comments and useful discussions. I am also grateful to the referees for their suggestions that helped improve this paper.

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  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    Hermie Monterde

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  1. Hermie Monterde
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Correspondence to Hermie Monterde.

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Monterde, H. Sedentariness in quantum walks. Quantum Inf Process 22, 273 (2023). https://doi.org/10.1007/s11128-023-04011-3

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