Impediments to diffusion in quantum graphs: Geometry-based upper bounds on the spectral gap

Berkolaiko, Gregory; Kennedy, James; Kurasov, Pavel; Mugnolo, Delio

doi:10.1090/proc/16322

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.

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