A note on self-adjoint extensions of the Laplacian on weighted graphs

By Xueping Huang, Matthias Keller, J. Masamune, Radoslaw K. Wojciechowski. 2012

119 citations recorded by the source corpus.

Abstract We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3b1b585c559621020008a5d950abc2227b293cef

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory (2010, unknown): “Among these we mention [1,2,15–18,25,34].”
Graphs of finite measure (2013, background): “Further studies involving intrinsic metrics on graphs can now be found in various references, including [2, 3, 21, 26].”
On the conservativeness and the recurrence of symmetric jump-diffusions (2012, background): “Furthermore, since B(R) ∩ V is finite, by the Hopf-Rinow type property of locally finite graphs [21], we are able to do this modification for any x ∈ B(R)∩V and any e ∈ E with x ∼ e.”
Cheeger inequalities for unbounded graph Laplacians (2012, background): “[14, 15, 16, 23, 25, 30, 31, 32, 42], where certain variations of the concept also go under the name of adapted metrics.”