Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs

Shi, Yongjie; Yu, Chengjie

doi:10.1090/proc/15866

Comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs
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by Yongjie Shi and Chengjie Yu PDF

Proc. Amer. Math. Soc. 150 (2022), 1505-1517 Request permission

Abstract:

In this paper, we obtain a comparison of Steklov eigenvalues and Laplacian eigenvalues on graphs and discuss its rigidity. As applications of the comparison of eigenvalues, we obtain Lichnerowicz-type estimates and some combinatorial estimates for Steklov eigenvalues on graphs.

References

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
Andries E. Brouwer and Willem H. Haemers, Spectra of graphs, Universitext, Springer, New York, 2012. MR 2882891, DOI 10.1007/978-1-4614-1939-6
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, Minimum higher eigenvalues of Laplacians on graphs, Duke Math. J. 83 (1996), no. 1, 1–18. MR 1388841, DOI 10.1215/S0012-7094-96-08301-5
Miroslav Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23(98) (1973), 298–305. MR 318007, DOI 10.21136/CMJ.1973.101168
W. Han and B. Hua, Steklov eigenvalue problem on subgraphs of integer lattics, arXiv:1902.05831, 2019.
Asma Hassannezhad and Laurent Miclo, Higher order Cheeger inequalities for Steklov eigenvalues, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 1, 43–88 (English, with English and French summaries). MR 4093440, DOI 10.24033/asens.2417
Bobo Hua, Yan Huang, and Zuoqin Wang, First eigenvalue estimates of Dirichlet-to-Neumann operators on graphs, Calc. Var. Partial Differential Equations 56 (2017), no. 6, Paper No. 178, 21. MR 3722072, DOI 10.1007/s00526-017-1260-3
B. Hua, Y. Huang, and Z. Wang, Cheeger esitmates o Dirichlet-to-Neumann operators on infinite subgraphs of graphs, arXiv:1810.10763, 2018.
Nikolay Kuznetsov, Tadeusz Kulczycki, Mateusz Kwaśnicki, Alexander Nazarov, Sergey Poborchi, Iosif Polterovich, and Bartłomiej Siudeja, The legacy of Vladimir Andreevich Steklov, Notices Amer. Math. Soc. 61 (2014), no. 1, 9–22. MR 3137253, DOI 10.1090/noti1073
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
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
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
Hélène Perrin, Isoperimetric upper bound for the first eigenvalue of discrete Steklov problems, J. Geom. Anal. 31 (2021), no. 8, 8144–8155. MR 4293927, DOI 10.1007/s12220-020-00572-2
Y. Shi and C. Yu, Higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues, arXiv:1904.03880, 2019.
Y. Shi and C. Yu, A Lichnerowicz-type estimate for Steklov eigenvalues on graphs and its rigidity, Preprint, arXiv:2010.13966, 2020.
W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. École Norm. Sup. (3) 19 (1902), 191–259 (French). MR 1509012, DOI 10.24033/asens.510
L. Tschanz, Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs, arXiv:2101.04402, 2021.
C. Xia and C. Xiong, Escobar’s conjecture on a sharp lower bound for the first nonzero steklov eigenvalue, arXiv:1907.07340, 2019.