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Lower bounds for the first eigenvalue of the Steklov problem on graphs

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Abstract

We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal of the boundary is 2, and asymptotically sharp as the diameter of the boundary tends to infinity in the other cases. We also investigate the case of weighted graphs and compare our result to the Cheeger inequality.

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Acknowledgements

I would like to thank the anonymous referee for his/her helpful comments and suggestions. I would like to thank Professors Bruno Colbois and Alain Valette for the helpful discussions about the manuscript.

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  1. Institut de mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000, Neuchâtel, Switzerland

    Hélène Perrin

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  1. Hélène Perrin
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Correspondence to Hélène Perrin.

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Communicated by J. Jost.

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Perrin, H. Lower bounds for the first eigenvalue of the Steklov problem on graphs. Calc. Var. 58, 67 (2019). https://doi.org/10.1007/s00526-019-1516-1

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