TY  - JOUR
A1  - Liu, Shiping
A1  - Münch, Florentin
A1  - Peyerimhoff, Norbert
T1  - Bakry-Emery curvature and diameter bounds on graphs
JF  - Calculus of variations and partial differential equations
N2  - We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Émery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet–Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672–698, 2018) and Horn et al. (J für die reine und angewandte Mathematik (Crelle’s J), 2017, https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry–Émery curvature functions of graphs, 2016).
Y1  - 2018
U6  - https://doi.org/10.1007/s00526-018-1334-x
SN  - 0944-2669
SN  - 1432-0835
VL  - 57
IS  - 2
PB  - Springer
CY  - Heidelberg
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Liu, Shiping
A1  - Peyerimhoff, Norbert
T1  - A note on eigenvalue bounds for non-compact manifolds
JF  - Mathematische Nachrichten
N2  - In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.
KW  - Cheeger inequality
KW  - eigenvalues
KW  - Laplacian
KW  - negative curvature
KW  - Riemannian manifold
Y1  - 2021
U6  - https://doi.org/10.1002/mana.201900209
SN  - 0025-584X
SN  - 1522-2616
VL  - 294
IS  - 6
SP  - 1134
EP  - 1139
PB  - Wiley-VCH
CY  - Weinheim
ER  -