Abstract
A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we determine the spectrum as a set via solutions, discuss upper and lower spectral bounds by isoperimetric constants and volume growth and study p-independence of spectra under a volume growth assumption.
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Acknowledgements
The author expresses his gratitude to Daniel Lenz for introducing him to the guiding theme of this survey that intrinsic metrics are the answer to many questions. Furthermore, several inspiring discussions with Xueping Huang resulted in various insights that improved this article. Moreover, the author appreciates the support and the kind invitation to the conference on ‘Mathematical Technology of Networks—QGraphs 2013’ at the ZiF in Bielefeld by Delio Mugnolo. Finally, funding by the DFG is acknowledged.
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Keller, M. (2015). Intrinsic Metrics on Graphs: A Survey. In: Mugnolo, D. (eds) Mathematical Technology of Networks. Springer Proceedings in Mathematics & Statistics, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-16619-3_7
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