Laplacians on Infinite Graphs
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Beschreibung
The main focus in this memoir is on Laplacians on both weighted graphs and weighted
metric graphs. Let us emphasize that we consider infinite locally finite graphs and do not make any further
geometric assumptions. Whereas the existing literature usually treats these two types of Laplacian operators
separately, we approach them in a uniform manner in the present work and put particular emphasis on the
relationship between them. One of our main conceptual messages is that these two settings should be regarded
as complementary (rather than opposite) and exactly their interplay leads to important further insight
on both sides. Our central goal is twofold. First of all, we explore the relationships between these two objects
by comparing their basic spectral (self-adjointness, spectral gap, etc.), parabolic (Markovian uniqueness,
recurrence, stochastic completeness, etc.), and metric (quasi isometries, intrinsic metrics, etc.) properties. In
turn, we exploit these connections either to prove new results for Laplacians on metric graphs or to provide
new proofs and perspective on the recent progress in weighted graph Laplacians. We also demonstrate our
findings by considering several important classes of graphs (Cayley graphs, tessellations, and antitrees).
Autor*in
Aleksey Kostenko
University of Ljubljana, Slovenia;
and University of Vienna, Austria
Themen in »Laplacians on Infinite Graphs«
Graph metric graph Laplacians on graphs spectral graph theory self-adjointness Markovian uniqueness spectral gap recurrence ultracontractivity