Comparison of the Dirichlet and Laplacian eigenvalues on graphs and its rigidity
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- by Yongjie Shi and Chengjie Yu;
- Proc. Amer. Math. Soc.
- DOI: https://doi.org/10.1090/proc/17289
- Published electronically: June 27, 2025
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Abstract:
In this paper, we obtain a comparison of the Dirichlet eigenvalues and Laplacian eigenvalues on weighted graphs with boundary and discuss its rigidity.References
- Martin T. Barlow, Random walks and heat kernels on graphs, London Mathematical Society Lecture Note Series, vol. 438, Cambridge University Press, Cambridge, 2017. MR 3616731, DOI 10.1017/9781107415690
- F. Bauer, F. Chung, Y. Lin, and Y. Liu, Curvature aspects of graphs, Proc. Amer. Math. Soc. 145 (2017), no. 5, 2033–2042. MR 3611318, DOI 10.1090/proc/13145
- Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1421568
- José F. Escobar, An isoperimetric inequality and the first Steklov eigenvalue, J. Funct. Anal. 165 (1999), no. 1, 101–116. MR 1696453, DOI 10.1006/jfan.1999.3402
- Joel Friedman, Some geometric aspects of graphs and their eigenfunctions, Duke Math. J. 69 (1993), no. 3, 487–525. MR 1208809, DOI 10.1215/S0012-7094-93-06921-9
- Miroslav Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23(98) (1973), 298–305. MR 318007
- Alexander Grigor′yan, Introduction to analysis on graphs, University Lecture Series, vol. 71, American Mathematical Society, Providence, RI, 2018. MR 3822363, DOI 10.1090/ulect/071
- Bobo Hua, Yan Huang, and Zuoqin Wang, Cheeger estimates of Dirichlet-to-Neumann operators on infinite subgraphs of graphs, J. Spectr. Theory 12 (2022), no. 3, 1079–1108. MR 4578345, DOI 10.4171/jst/427
- Bo’az Klartag, Gady Kozma, Peter Ralli, and Prasad Tetali, Discrete curvature and abelian groups, Canad. J. Math. 68 (2016), no. 3, 655–674. MR 3492631, DOI 10.4153/CJM-2015-046-8
- André Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, III, Dunod, Paris, 1958 (French). MR 124009
- Yong Lin, Linyuan Lu, and Shing-Tung Yau, Ricci curvature of graphs, Tohoku Math. J. (2) 63 (2011), no. 4, 605–627. MR 2872958, DOI 10.2748/tmj/1325886283
- Florentin Münch and Radosław K. Wojciechowski, Ollivier Ricci curvature for general graph Laplacians: heat equation, Laplacian comparison, non-explosion and diameter bounds, Adv. Math. 356 (2019), 106759, 45. MR 3998765, DOI 10.1016/j.aim.2019.106759
- Robert C. Reilly, Applications of the integral of an invariant of the Hessian, Bull. Amer. Math. Soc. 82 (1976), no. 4, 579–580. MR 415540, DOI 10.1090/S0002-9904-1976-14110-9
- Hélène Perrin, Lower bounds for the first eigenvalue of the Steklov problem on graphs, Calc. Var. Partial Differential Equations 58 (2019), no. 2, Paper No. 67, 12. MR 3925558, DOI 10.1007/s00526-019-1516-1
- Yongjie Shi and Chengjie Yu, Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs, Proc. Amer. Math. Soc. 150 (2022), no. 4, 1505–1517. MR 4375740, DOI 10.1090/proc/15866
- Yongjie Shi and Chengjie Yu, A Lichnerowicz-type estimate for Steklov eigenvalues on graphs and its rigidity, Calc. Var. Partial Differential Equations 61 (2022), no. 3, Paper No. 98, 22. MR 4400618, DOI 10.1007/s00526-022-02213-8
- Yongjie Shi and Chengjie Yu, Comparison of Dirichlet, Neumann and Laplacian Eigenvalues on Graphs and Applications, arXiv:2011.04160v4, 2024, Submitted.
- Chao Xia and Changwei Xiong, Escobar’s conjecture on a sharp lower bound for the first nonzero Steklov eigenvalue, Peking Math. J. 7 (2024), no. 2, 759–778. MR 4792989, DOI 10.1007/s42543-023-00068-2
Bibliographic Information
- Yongjie Shi
- Affiliation: Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
- Email: yjshi@stu.edu.cn
- Chengjie Yu
- Affiliation: Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
- Email: cjyu@stu.edu.cn
- Received by editor(s): October 20, 2024
- Received by editor(s) in revised form: October 24, 2024, March 10, 2025, and March 16, 2025
- Published electronically: June 27, 2025
- Additional Notes: The first author’s research was partially supported by NNSF of China with contract no. 11701355.
The second author’s research was partially supported by GDNSF with contract no. 2025A1515011144 and NNSF of China with contract no. 11571215. - Communicated by: Nageswari Shanmugalingam
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
- MSC (2020): Primary 05C50; Secondary 39A12
- DOI: https://doi.org/10.1090/proc/17289