Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

By Matthias Keller, Daniel Lenz. 2011

159 citations recorded by the source corpus.

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/615ba0dc5533b7604dd141e15620722d35116a6f

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Citations in the corpus

Cheeger inequalities for unbounded graph Laplacians (2012, background): “For example, they are already found in [37] (see also [22]).”
Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory (2010, background): “For a further detailed study of various (spectral) aspects of this Dirichlet form we refer to [12, 14, 15].”
Graphs of finite measure (2013, background): “ntial theory of inﬁnite graphs has been a ﬂourishing subject in recent years. In particular, the case of inﬁnite graphs with possibly unbounded vertex degree has received substantial attention, e.g., [1, 7, 14, 15, 16, 20, 25, 26, 27, 28, 29, 30, 33, 34, 36, 43, 50, 54, 55] and references therein. I”
Generalized Solutions and Spectrum for Dirichlet Forms on Graphs (2010, background): “Moreover in this case the restriction of L̃ is bounded on (V,m) for all p ∈ [1,∞] (for details see Section 3 of [13]).”
A note on self-adjoint extensions of the Laplacian on weighted graphs (2012, background): “The following auxiliary lemmas are well known in various other situations, see [11, 22, 23, 27, 28, 30, 31].”
Stochastic completeness for graphs with curvature dimension conditions (2015, background): “n is called combinatorial (or physical) Laplacian. Note that normalized Laplacians are boundedoperators, sothat these graphs are always stochastically complete, see Dodziuk [Dod06] or Keller and Lenz [KL10]. Thus, the only interesting cases are combinatorial Laplacians, or more general unbounded Lap”
Dirichlet forms and stochastic completeness of graphs and subgraphs (2009, unknown): “For further studies of certain spectral features of Laplacians in the framework developed below we refer the reader to [16], [19], both of which were written after the present paper.”