Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap
arXiv:2206.10046 · doi:10.1090/proc/16322
Abstract
We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. 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.
arXiv comment15+epsilon pages, 4 figures; with helpful corrections from an anonymous referee
References in corpus (9)
- On the Spectral Gap of a Quantum Graph
- Surgery principles for the spectral analysis of quantum graphs
- Gaussian estimates for a heat equation on a network
- Spectral gap for quantum graphs and their connectivity
- Brownian Motions on Metric Graphs
- Quantum graphs which optimize the spectral gap
- Hearing Euler characteristic of graphs
- Spectral Estimates for Infinite Quantum Graphs
- Three dimensional elastic beam frames: rigid joint conditions in variational and differential formulation