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  -