Impediments to diffusion in quantum graphs: Geometry-based upper bounds on the spectral gap
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- by Gregory Berkolaiko, James B. Kennedy, Pavel Kurasov and Delio Mugnolo;
- Proc. Amer. Math. Soc. 151 (2023), 3439-3455
- DOI: https://doi.org/10.1090/proc/16322
- Published electronically: April 28, 2023
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Abstract:
We derive several upper bounds on the spectral gap of the Laplacian on compact metric graphs with standard or Dirichlet vertex conditions. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling.References
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Bibliographic Information
- Gregory Berkolaiko
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 366171
- ORCID: 0000-0002-3335-8901
- Email: gberkolaiko@tamu.edu
- James B. Kennedy
- Affiliation: Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, P-1749-016 Lisboa, Portugal
- MR Author ID: 829307
- ORCID: 0000-0001-5634-0301
- Email: jbkennedy@fc.ul.pt
- Pavel Kurasov
- Affiliation: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden
- MR Author ID: 265224
- ORCID: 0000-0003-3256-6968
- Email: kurasov@math.su.se
- Delio Mugnolo
- Affiliation: Lehrgebiet Analysis, Fakultät Mathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany
- MR Author ID: 712380
- ORCID: setImmediate$0.49781310504760423$4
- Email: delio.mugnolo@fernuni-hagen.de
- Received by editor(s): June 22, 2022
- Received by editor(s) in revised form: September 13, 2022, September 30, 2022, and October 10, 2022
- Published electronically: April 28, 2023
- Additional Notes: The work of the first author was partially supported by the NSF under grant DMS–1815075.
The second author was supported by the Fundação para a Ciência e a Tecnologia, Portugal, via the program “Investigador FCT”, reference IF/01461/2015, and project PTDC/MAT-PUR/1788/2020.
The third author was partially supported by the Swedish Research Council (Grant 2020-03780).
The fourth author was partially supported by the Deutsche Forschungsgemeinschaft (Grant 397230547). All four authors were partially supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld, Germany, within the framework of the cooperation group on “Discrete and continuous models in the theory of networks”.
The contribution of the last three authors is based upon work from COST Action 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu. - Communicated by: Tanya Christiansen
- © Copyright 2023 by the authors
- Journal: Proc. Amer. Math. Soc. 151 (2023), 3439-3455
- MSC (2020): Primary 34B45; Secondary 05C50, 35P15, 81Q35
- DOI: https://doi.org/10.1090/proc/16322
- MathSciNet review: 4591778