Abstract
In this paper, we obtain some comparisons of the Dirichlet, Neumann and Laplacian eigenvalues on graphs. We also discuss their rigidities and some of their applications including some Lichnerowicz-type, Fiedler-type and Friedman-type estimates for Dirichlet eigenvalues and Neumann eigenvalues. The comparisons on Neumann eigenvalues can be translated as comparisons on Steklov eigenvalues in our setting. So, some of the results can be viewed as extensions for parts of [11] by Hua-Huang-Wang, and parts of our previous works [20, 21].
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Acknowledgements
The authors would like to thank the referee for helpful comments which reduce the length of the paper considerably, and informing us that Perrin [17] has studied the relevance of the Steklov problem on graphs with different possible definitions of boundaries and shown that results on the most general definition of graphs with boundary can be related with results for a more restrictive definition.
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Y. Shi: Research partially supported by NNSF of China with contract no. 11701355.
C. Yu: Research partially supported by GDNSF with contract no. 2025A1515011144 and NNSF of China with contract no. 11571215.
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Shi, Y., Yu, C. Comparisons of Dirichlet, Neumann and Laplacian eigenvalues on graphs and their applications. Calc. Var. 64, 305 (2025). https://doi.org/10.1007/s00526-025-03178-0
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DOI: https://doi.org/10.1007/s00526-025-03178-0