TY  - JOUR
A1  - Keller, Matthias
A1  - Lenz, Daniel
A1  - Münch, Florentin
A1  - Schmidt, Marcel
A1  - Telcs, Andras
T1  - Note on short-time behavior of semigroups associated to self-adjoint operators
JF  - Bulletin of the London Mathematical Society
N2  - We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times t roughly like t(d), where d is the combinatorial distance. This is very different from the classical Varadhan-type behavior on manifolds. Moreover, this also gives that short-time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.
Y1  - 2016
U6  - https://doi.org/10.1112/blms/bdw054
SN  - 0024-6093
SN  - 1469-2120
VL  - 48
SP  - 935
EP  - 944
PB  - Oxford Univ. Press
CY  - Oxford
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Münch, Florentin
A1  - Pogorzelski, Felix
T1  - Geometry and spectrum of rapidly branching graphs
JF  - Mathematische Nachrichten
N2  - We study graphs whose vertex degree tends to infinity and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
KW  - Graph Laplacians
KW  - discrete spectrum
KW  - eigenvalue asymptotics
KW  - isoperimetric estimates
KW  - stochastic completeness
Y1  - 2016
U6  - https://doi.org/10.1002/mana.201400349
SN  - 0025-584X
SN  - 1522-2616
VL  - 289
SP  - 1636
EP  - 1647
PB  - Wiley-VCH
CY  - Weinheim
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Mugnolo, Delio
T1  - General Cheeger inequalities for p-Laplacians on graphs
JF  - Theoretical ecology
N2  - We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which uses the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of p-Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls. (C) 2016 Elsevier Ltd. All rights reserved.
KW  - Cheeger inequalities
KW  - Spectral theory of graphs
KW  - Intrinsic metrics for Dirichlet forms
Y1  - 2016
U6  - https://doi.org/10.1016/j.na.2016.07.011
SN  - 0362-546X
SN  - 1873-5215
VL  - 147
SP  - 80
EP  - 95
PB  - Elsevier
CY  - Oxford
ER  -